Time dispersion in quantum mechanics

John Ashmead

June 4, 2018

“Wheeler’s often unconventional vision of nature was grounded in re-ality through the principle of radical conservatism, which he acquiredfrom Niels Bohr: Be conservative by sticking to well-establishedphysical principles, but probe them by exposing their most radicalconclusions.” – Kip Thorne [80].

“You can have as much junk in the guess as you like, provided thatthe consequences can be compared with experiment.” – RichardFeynman [27]

Abstract

In quantum mechanics the time dimension is treated as a parame-ter, while the three space dimensions are treated as observables. Thisassumption is both untested and inconsistent with relativity.

From dimensional analysis, we expect quantum effects along the timeaxis to be of order an attosecond. Such effects are not ruled out by cur-rent experiments. But they are large enough to be detected with currenttechnology, if sufficiently specific predictions can be made.

To supply such we use path integrals. The only change required is togeneralize the usual three dimensional paths to four. We treat the singleparticle case first, then extend to quantum electrodynamics.

We predict a large variety of testable effects. The principal effectsare additional dispersion in time and full equivalence of the time/energyuncertainty principle to the space/momentum one. Additional effects in-clude interference, diffraction, resonance in time, and so on.

Further the usual problems with ultraviolet divergences in QED dis-appear. We can recover them by letting the dispersion in time go to zero.As it does, the uncertainty in energy becomes infinite – and this in turnmakes the loop integrals diverge. It appears it is precisely the assumptionthat quantum mechanics does not apply along the time dimension thatcreates the ultraviolet divergences.

The approach here has no free parameters; it is therefore falsifiable.As it treats time and space with complete symmetry and does not sufferfrom the ultraviolet divergences, it may provide a useful starting point forattacks on quantum gravity.

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Contents1 Introduction 3

1.1 Extension of the wave function in time . . . . . . . . . . . . 31.2 Estimated scale of effects . . . . . . . . . . . . . . . . . . . 71.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Path integrals 102.1 Laboratory time . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Coordinate time . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Initial wave function . . . . . . . . . . . . . . . . . . . . . . 122.4 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Convergence and normalization of the path integral . . . . . 152.6 Formal expression for the path integral . . . . . . . . . . . . 19

3 Derivation of the Schrödinger equation 19

4 Long, slow approximation 214.1 Fixing the scale and additive constants . . . . . . . . . . . . 214.2 Physical meaning of the long, slow approximation . . . . . . 224.3 Can we make an observer independent choice of frame? . . 23

5 Free solutions 245.1 Physically reasonable definition of the initial wave function 245.2 Evolution of the free wave function in laboratory time . . . 295.3 Time of arrival measurements . . . . . . . . . . . . . . . . . 32

6 Semi-classical approximation 386.1 Derivation of the semi-classical approximation . . . . . . . . 386.2 Derivation of the free kernel using the semi-classical ap-

proximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3 Constant magnetic field . . . . . . . . . . . . . . . . . . . . 406.4 Constant electric field . . . . . . . . . . . . . . . . . . . . . 41

7 Single and double slit experiments 437.1 Single slit in time in standard quantum mechanics . . . . . 437.2 Single slit in time with quantization in time . . . . . . . . . 477.3 Double slit in time . . . . . . . . . . . . . . . . . . . . . . . 49

8 Multiple particles 508.1 Why look at the multiple particle case? . . . . . . . . . . . 508.2 ABC model . . . . . . . . . . . . . . . . . . . . . . . . . . . 528.3 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528.4 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.5 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608.6 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . 618.7 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . 638.8 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648.9 Point correction to the mass . . . . . . . . . . . . . . . . . . 658.10 Loop correction to the mass . . . . . . . . . . . . . . . . . . 68

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9 Discussion 729.1 Is TQ falsifiable? . . . . . . . . . . . . . . . . . . . . . . . . 739.2 Experimental effects . . . . . . . . . . . . . . . . . . . . . . 749.3 Extensions of TQ . . . . . . . . . . . . . . . . . . . . . . . . 749.4 Alternatives to TQ . . . . . . . . . . . . . . . . . . . . . . . 759.5 If TQ is falsified? . . . . . . . . . . . . . . . . . . . . . . . . 76

A Conventions 76A.1 Time and space variables . . . . . . . . . . . . . . . . . . . 76A.2 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . 77A.3 Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . 77

B Classical equations of motion 77

C Unitarity 78

D Gauge transformations 79

E Free wave functions and kernels 80E.1 Plane wave functions . . . . . . . . . . . . . . . . . . . . . . 80E.2 Gaussian wave functions . . . . . . . . . . . . . . . . . . . . 81E.3 Free kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

F Acknowledgments 87

List of Figures1.1 Wave functions for Alice and Bob . . . . . . . . . . . . . . . . . . 42.1 Morlet wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.1 Time of arrival in SQM . . . . . . . . . . . . . . . . . . . . . . . 335.2 Time of arrival in TQ . . . . . . . . . . . . . . . . . . . . . . . . 347.1 Single slit in time in standard quantum mechanics . . . . . . . . 437.2 Single slit in time with quantization in time . . . . . . . . . . . . 477.3 Double slit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.1 Vertex interactions in the AB model . . . . . . . . . . . . . . . . 528.2 Point correction to the mass . . . . . . . . . . . . . . . . . . . . . 668.3 Loop correction to the mass . . . . . . . . . . . . . . . . . . . . . 68

1 Introduction

1.1 Extension of the wave function in timeIn relativity, time and space enter on a basis of formal equivalence. In specialrelativity, the time and space coordinates rotate into each other under Lorentztransformations. In general relativity, the time and the radial coordinate changeplaces at the Schwarzschild radius (for instance in [4]). In wormholes and otherexotic solutions to general relativity, time can even curve back on itself [33, 79].

3

But in quantum mechanics “time is a parameter not an operator” (Hilgevoord[37, 38]). This is clear in the Schrödinger equation:

ıd

dτψτ (~x) = Hψτ (~x) (1.1)

Here the wave function is indexed by time: if we know the wave function at timeτ we can use this equation to compute the wave function at time τ + ε. Thewave function has in general non-zero dispersion in space, but is always treatedas having zero dispersion in time. This is not manifestly consistent with specialrelativity.

Figure 1.1: Wave functions for Alice and Bob

Consider Alice in laboratory, her co-worker Bob jetting around like a fusionpowered mosquito. Both are studying the same system but with respective wavefunctions:

ψ(Alice) (tAlice, xAlice)

ψ(Bob) (tBob, xBob)(1.2)

Alice and Bob center their respective wave functions on the particle:

〈tAlice〉 = 〈tBob〉 = 0

〈xAlice〉 = 〈xBob〉 = 0(1.3)

These are distinct wave functions but must give the same predictions for allobservables. But at Alice’s time zero, Bob’s wave function extends into her pastand future. And at Bob’s time zero her wave function extends into his past andfuture.

Given that the quantum wave function gives an extraordinarily accuratepicture of reality, this is at a minimum a bit strange. In the case of the three

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space dimensions, there is no problem rotating between coordinate systems, butin the case of four dimensions, this simplicity is lost1.

If both Alice and Bob were using fully four dimensional wave functions –wave functions that extended in time as well as space – then their wave functionswould be related by the associated Lorentz transformation. If Bob and Alice’scoordinate systems are related by a Lorentz transformation Λ:

x(Bob) = Λx(Alice) (1.4)

then their wave functions would be related by:

ψ(Bob)(x(Bob)

)= ψ(Alice)

(Λx(Alice)

)(1.5)

just as they would be for rotations in three space.This then is our basic hypothesis: the quantum mechanical wave function

should be extended in the time direction on the same basis as it is extendedalong the three space dimensions:

ψ (~x)→ ψ (t, ~x) (1.6)

This implies the wave function normally extends slightly into future and intopast. There are two principle effects:

1. Dispersion in time on same basis as dispersion in space. Physical wavefunctions are always a bit spread out in space; they will now also be a bitspread out in time.

2. The uncertainty principle for time/energy is treated on same basis as theuncertainty principle for space/momentum. If a particle’s position in timeis well-defined, its energy will highly uncertain and vice versa.

1.1.1 Dispersion in time

If the wave functions normally have an extension in time then every time-specificmeasurement should show additional dispersion in time.

Suppose we are measuring the time-of-arrival of a particle at a detector.Define the average time-of-arrival as:

⟨τ (TOA)

⟩≡∞∫−∞

dττp(TOA) (τ) (1.7)

with an associated uncertainty:

⟨∆τ (TOA)

⟩≡

√√√√√ ∞∫−∞

dττ2p(TOA) (τ)−⟨τ (TOA)

⟩2 (1.8)

1For a more technical and precise discussion of some of the difficulties see the introductionto Horwitz [39].

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The probability distribution for the particle will normally be spread out inspace, so its arrival times will also be spread out, depending on the velocity ofthe particle and its dispersion in space.

But if it also has a dispersion in time, then part of the wave function willreach the detector – thanks to the fuzziness in time – a bit sooner and also abit later than otherwise expected. There will be an additional dispersion in thetime-of-arrival due to the dispersion in time:

⟨∆τ (TOA)

⟩(wave function extended in time)

>⟨

∆τ (TOA)⟩(wave function not extended in time)

As we will see (below on page 32), at non-relativistic speeds, the dispersionin the time-of-arrival is dominated by the dispersion in space, so this effect maybe hard to pick out. At relativistic speeds, the contributions of the space andtime dispersions can be comparable.

1.1.2 Uncertainty principle for time and energy

Of particular significance for this work are differences in the treatment of theuncertainty principle for time/energy as opposed to that for space/momentum.

In the early days of quantum mechanics, these were treated on same basis.See for instance the discussions between Bohr and Einstein of the famous clock-in-a-box experiment [75] or the comments of Heisenberg in [36].

In later work this symmetry was lost. As Busch [16] puts it “. . . differenttypes of time energy uncertainty can indeed be deduced in specific contexts, but. . . there is no unique universal relation that could stand on equal footing withthe position-momentum uncertainty relation.” See also Pauli, Dirac, and Muga[66, 23, 64, 62].

There does not appear to have been any experimental test or observationalevidence for this; it is merely the way the field has developed.

That is not to say that there are not uncertainties with respect to time, butthey are side effects of other uncertainties in quantum mechanics. For instance,if a particle is moving to the right, spread out in space, and going towards adetector at a fixed position, its time-of-arrival will have a dispersion in time.But this is a side-effect of the dispersion in space.

So consider a particle going thru a narrow slit in time, as a camera shutter.Its wave function will be clipped in time. If the wave function is not extendedin time, then the wave function will merely be clipped: the resulting dispersionin time at detector will be reduced.

But if the wave function is extended in time and the Heisenberg uncertaintyprinciple applies in time/energy on the same basis as with space/momentum,then a very fast camera shutter will give a small uncertainty in time at the gate:

∆t→ 0 (1.9)

causing the uncertainty in energy to become arbitrarily great:

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∆E >1

∆t⇒ ∆E →∞ (1.10)

which will in turn cause the wave function to fan out in time and the dis-persion in time-of-arrival to become arbitrarily great:⟨

∆τ (TOA)⟩(wave function extended in time)

→∞ (1.11)

1.1.3 A necessary hypothesis

This question does not appear to have been attacked directly. As noted, the as-sumption that the wave function is not extended in time seems to have crept intothe literature of its own, without experimental test or observational evidence.

To make an experimental test of this question we have to develop predictionsfor both branches:

1. Assume the wave function is not extended in time. Make predictions abouttime-of-arrival and the like.

2. Assume the wave function is extended in time. Make equivalent predic-tions.

3. Compare.

Note we have to develop both branches in a way that makes the comparisonstraightforward. We will prefer simplicity to rigor.

Further, to make the results falsifiable we have to develop the extended intime branch in a way that is clearly correct. A null result should show that thewave function is not extended in time.

These twin requirements will motivate much of what follows.

1.2 Estimated scale of effectsHas this hypothesis has already been falsified? Quantum mechanics has beentested with extraordinary precision. Should associated effects have been seenalready, even if not looked for?

Consider the atomic scale given by the Bohr radius 5.3 10−11m. We takethis as an estimate of the uncertainty in space.

We assume the maximum symmetry possible between time and space. Wetherefore infer that the uncertainty in time should be of order the uncertaintyin space (in units where c = 1).

Dividing by the speed of light we get the Bohr radius in time a0 = .177 10−18s,or less than an attosecond. .177as is therefore our starting estimate of the un-certainty in time. In natural units, which we will be using here unless otherwisestated, this is a0 = 2.68x10−4eV −1 or (αm)

−1 where α is the fine structure

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constant and m the mass of the electron. Therefore from strictly dimensionaland symmetry arguments, the effects will be small, of order attoseconds2.

This is sufficient to explain why such effects have not been seen.At the same time, the time scales we can look at experimentally are now

getting down to the attosecond range. A recent paper by Ossiander et al [65]reports results at the sub-attosecond level.

Therefore if we can provide the experimentalists with a sufficiently well-defined target, the hypothesis should be falsifiable in practice.

1.3 Organization“In other words, we are trying to prove ourselves wrong as quickly aspossible, because only in that way can we find progress.” – Feynman[27]

Our goal is to establish experimental tests that will allow us to falsify the hy-pothesis that the quantum mechanical wave function should be extended inthe time direction on the same basis as it is extended along the three spacedimensions.

Our starting requirements are:

1. The most complete possible equivalence in the treatment of time and space– manifest covariance at every point at a minimum.

2. Consistency with existing results.

These two conditions in combination leave us – as we will see – with no freeparameters. And having no free parameters means in turn that our hypothesisis falsifiable in principle.

To get to falsifiable in practice, we will look for the simplest cases that makea direct comparison possible. We will also look at points of principle that thatneed to be addressed, particularly if they might have implications that falsifyour hypothesis before it gets to an experimental test (as violations of unitarityor inability to renormalize).

Since the phrase “the quantum mechanical wave function should be extendedin the time direction on the same basis as it is extended along the three spacedimensions” is a bit cumbersome, we will refer to our hypothesis as TQ ortemporal quantization.

We will refer to standard quantum mechanics or the null hypothesis as SQMor standard quantum mechanics.

It should be clear we see TQ as really SQM, just pushed a bit harder in thetime direction.

And we do not mean by “temporal quantization” that time itself is quan-tized! since we are extruding SQM into time, and SQM treats the three spacedimensions as continuous, we are treating time as continuous as well.

2But still 26 orders of magnitude greater than the Planck time tPlanck ≡√

~Gc5 ≈

5.39116x10−44s = 5.39116x10−26as

8

We will address first the single particle case, then the multiple (field theory).

1.3.1 Single particle case

To develop TQ in the single particle case we start with the familiar path integralapproach. Here we work by generalizing the usual paths in space to be paths intime and space. We use Morlet wavelet decomposition to define the initial wavefunctions and to ensure convergence of the path integrals. We use a Lagrangianthat conveniently works for both the SQM and TQ cases (up to additive andscale constants). We leave the rest of the path integral machinery untouched.

This gives us an expression for the path integral kernel for a charged par-ticle in an electromagnetic field. We derive the TQ version of the Schrödingerequation by taking the short time limit. We argue that the long time limit ofTQ should give SQM. And this in turn fixes the additive and scale constants inthe Lagrangian.

With this we have enough machinery to look at the free particle case, com-paring time-of-arrival dispersions in TQ and SQM. We get the initial estimateof the wave functions by looking for the solutions that maximize total entropy.Evolving the wave functions to get to the detector is straightforward. We ap-proximate the arrival at the detector in a way that lets us treat TQ and SQMon the same basis.

The result is that for non-relativistic particles, the dispersion at the detectoris likely to be dominated by the existing dispersion in space. So that the effectsof dispersion in time from TQ might easily be lost in the error bars.

However the effects of the uncertainty principle in time/energy can be muchgreater. We look at the single slit experiment in time. We assume the particleis sent through a gate open for a very short period of time. In SQM a short timewindow reduces the dispersion in the time-of-arrival, but in TQ it increases it.The effect may be made arbitrarily large by making the gate shorter.

We also look at the double slit experiment. Here the interference effects thatcharacterize that are the same in TQ as in SQM, so the double slit experimentdoes not provide as decisive a comparison.

1.3.2 Multiple particle case

The effects of TQ will be strongest at relativistic velocities and short times(high energies). This implies we should look at extending TQ to field the-ory. Extending the single particle work to Feynman diagrams without loops isstraightforward – we can just replace the SQM propagators with TQ propaga-tors and calculate. But diagrams containing loops are potentially problematic:with one more dimension to integrate over, the loop diagrams could go frombarely renormalizable to not being renormalizable at all. This would definitelybe a counter-indication.

We therefore extend TQ to include the multiple particle case.To do this we add to our starting requirements we add a third: the prop-

agator for the multiple case should match the one for the single particle case.

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Then we work as in the single particle case: we extend the paths from three tofour dimensions, find a Lagrangian which satisfies the requirements, and thenfind approximations which let us make the most direct possible comparison ofTQ to SQM.

For our test case we use a simple field theory with three massive spinlessparticles, A, B, and C – corresponding loosely to electron, photon, and proton.We look at various simple cases to make sure the derived Feynman rules makesense and match the corresponding SQM results.

Interestingly, the familiar ultraviolet divergences do not appear. We usephysically reasonable wave functions to start, with bounded uncertainties intime and space. The time and space components in the loop integrals areentangled with the time and space components in the initial wave functions, soinherit their bounded character. As a result the loop integrals go from beingpolynomially infinite to be being finite by inspection.

In the limit as we assume the initial wave functions have zero dispersion intime (the SQM assumption) their dispersion in energy becomes infinite, causingthe loop integrals to again diverge. The implication is that it is precisely the im-plicit assumption in SQM that time is “crisp” that is introducing the ultravioletdivergences.

1.3.3 Overall

With this done, we will argue in the discussion:

1. that TQ is not ruled out a priori.

2. that TQ is falsifiable. And given experimental work like Ossiander’s, prob-ably with current technology.

3. that TQ is useful as a source of interesting experiments. With TQ, wecan get well-defined predictions for any quantum effect along the timedimension: dispersion, interference, diffraction, resonance, tunneling, andso on. TQ is to SQM – with respect to time – as SQM is to classicalmechanics with respect to space. All quantum effects currently seen shouldhave an “in time” analog.

4. that as TQ is by construction fully symmetric between time and space,and it is not troubled by the ultraviolet divergences, it is a reasonablestarting point for attacks on the problem of quantum gravity.

5. that as TQ is built by combining relativity and quantum mechanics in themost straightforward possible way, experiments that falsify TQ may haveimplications for quantum mechanics and relativity.

2 Path integralsWe take as our starting point Feynman’s path integral approach to quantummechanics ([28, 76, 70, 77, 46, 44, 35, 90, 48, 89]). The advantage is that with

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path integrals the only change we have to make is to generalize the paths fromthree to four dimensions. What exactly this means and how to do it will requirea bit of thought, however.

With the path integral approach if we are given the wave function ψ0 (x0) attime zero and wish to compute the wave function ψT (xT ) at time T we sum overall paths from start to finish, weighing each by the integral of the Lagrangian:

ψT (xT ) =

T∫0

Dxτ exp

ı T∫0

dτL [x, x]

ψ0 (x0) (2.1)

where the meanings of the various terms here will be made more precise aswe continue.

One of the advantages of the path integral approach is that it has a naturalconnection to the classical approach: the path given by the solution of theclassical equations of motion is in some sense the “most typical” path. Thequantum paths can be seen as variations around the classical path.

If Alice is walking her dog from her house to Bob’s, she will naturally take theshortest possible path, which is also the classical path. But her dog will exploreinteresting side paths around her path. If her dog is a properly quantum dog itwill simultaneously explore all paths around the classical path.

In SQM, the paths vary in the three space dimensions. The measure Dxτcounts each such path once. In TQ, the paths are allowed – by definition – tovary in time as well, so they can dart around a bit into the future or a bit intothe past with each step. For each specific path in SQM, there will be an infinitenumber of paths in TQ, taking all possible routes into past and future, providedthey start at time zero and finish at time T .

In SQM it is as if the dog can only go side to side, to Alice’s left and right.But in TQ, the dog can dart ahead and behind Alice. Or overshoot Bob’s house,or start out by going in the wrong direction, or even loop around her severaltimes. So long as Alice’s dog starts and finishes at the same time and place asAlice starts and finishes, it doesn’t matter which route he takes. In fact ourquantum dog is again required to take all routes in time as in space.

To actually compute the path integral we will need to look at what timemeans to Alice, to her dog, how to describe the initial wave function of her dog,the Lagrangian that weights each path, and how we actually do the sums, beforewe can compute the amplitude for Alice’s dog to arrive at Bob’s doorstep.

2.1 Laboratory timeIn classical mechanics when we look at the action, at the integral of the La-grangian over time:

T∫0

dτL [x, x] (2.2)

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we are free to take the parameter as any monotonically increasing variable.We will get the same classical equations of motion in any case.

But we specifically choose to use the time as shown on a laboratory clock. Wetake the term laboratory time from Busch [16]. We will refer to the laboratorytime as τ . We will use the terms clock time and laboratory time interchangeably.And of course it can be any agreed clock, if Alice is out walking her dog herdigital watch will do.

It is useful to visualize this clock as a metronome, breaking up the clocktime into a series of N ticks each of length ε. If τ = 0 at the source, and τ = Tat the detector, we have:

ε ≡ T

N(2.3)

We will take the limit as N →∞ as the final step in the calculation.

2.2 Coordinate timeNow we visualize a four dimensional coordinate system with the wave functionextended in all four dimensions. We will refer to t as coordinate time by anal-ogy to the three coordinate space dimensions: coordinate x, coordinate y, andcoordinate z.

Paths are defined with reference to this coordinate system. If the time byAlice’s watch is τ , then each path π will have a location at τ given by:

πτ (t, x, y, z) (2.4)

If we imagine the coordinate system treated as a series of discrete points,with indexes h, i, j, k then each path is defined by the series of values it has forh, i, j, k at each instant in clock time. In fact, the path integral measure Dx isoften defined by assigning a weight of one to each distinct path, and then takingthe limit as the spacing goes to zero.

2.3 Initial wave functionWe need a starting set of wave functions ψ0 at clock time τ = 0. We willneed wave functions that extend in both coordinate time and space. The usualchoices would be δ functions or plane waves.

In coordinate time these might be:

δ(t− t0) (2.5)

e−ıE(t−t0) (2.6)

Or in space:

δ(x− x) (2.7)

12

eıp(x−x0) (2.8)

But neither δ functions nor plane waves are physical. Their use creates arisk of artifacts.

More physical would be Gaussian test functions, for instance in coordinatetime:

ϕ (t) ≡ 4

√1

πσ2t

e−ıE(t−t0)− (t−t0)2

2σ2t (2.9)

Or in space:

ϕ (x) ≡ 4

√1

πσ2x

eıp(x−x0)− (x−x0)2

2σ2x (2.10)

But while Gaussian test functions are physically reasonable they are notcompletely general.

We can achieve both generality and physical reasonableness by using a basisof Morlet wavelets ([59, 17, 58, 42, 11, 14, 3, 82, 8, 9]).

-3 -2 -1 1 2 3

-0.4

-0.2

0.2

0.4

real part

imaginary part

Figure 2.1: Morlet wavelet

Morlet wavelets are derived by starting with a “mother” wavelet

φ(mother) (t) ≡(e−ıt − 1√

e

)e

(− t22

)(2.11)

and scaling and displacing it with the replacement t→ t−ls

φsl (t) =1√|s|

(e−ı(

t−ls ) − 1√

e

)e−

12 ( t−ls )

2

(2.12)

Any normalizable function f can be broken up into wavelet components f using

fsl =

∞∫−∞

dtφ∗sl (t) f (t) (2.13)

And then recovered using the inverse Morlet wavelet transform:

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f (t) =1

C

∞∫−∞

dsdl

s2φsl (t) fsl (2.14)

The value of C is worked out in [9].Therefore we can write any physically reasonable wave function in space as:

ψ (t) =1

C

∫dsdl

s2ψslφsl (t) (2.15)

And include space by using products of Morlet wavelets:

ψ (t, x) =1

C2

∫dstdlts2t

dsxdlxs2x

ψstltsxlxφstlt (x)φsxlx (x) (2.16)

Clearly this could rapidly become hopelessly cumbersome.Fortunately we do not need to actually perform the Morlet wavelet analyses:

we merely need the ability to do so. As each Morlet wavelet may be written as asum of a pair of Gaussians, Morlet wavelet analysis lets us write any physicallyreasonable wave function as a sum over Gaussians. Provided we are dealingonly with linear operations – the case throughout here – we can work directlywith Gaussian test functions. By Morlet wavelet analysis the results will thenbe valid for any physically reasonable wave functions.

2.4 LagrangianTo sum over the paths – to construct the path integral – we will need to weighteach path by the exponential of the action, where the action is defined as theintegral of the Lagrangian over the laboratory time:

eıB∫A

dτL(xµ,xµ)

(2.17)

We require a Lagrangian which:

1. Is manifestly covariant,

2. Produces the correct classical equations of motion,

3. And gives the correct Schrödinger equation.

We find an appropriate Lagrangian in Goldstein [34].

L (xµ, xµ) = −1

2mxµxµ − qxµAµ (x) (2.18)

This Lagrangian is unusual in that it uses four independent variables (the usualthree space coordinates plus a time variable) but still gives the familiar classicalequations of motion (see appendix B on page 77):

d

dτ

δL

δxµ− δL

δxµ= 0 (2.19)

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This Lagrangian therefor provides a natural bridge from a three to a four di-mensional picture.

The classical equations of motion are still produced if we add a dimensionlessscale a and an additive constant b to the Lagrangian:

− 1

2amxµxµ − aqxµAµ (x)− abm

2(2.20)

The requirement that we match the SQM results will fix a and b; see belowon page 21.

It is striking that we can find an acceptable Lagrangian that gives the sameclassical paths in both three dimensional and four dimensional contexts.

2.5 Convergence and normalization of the path integral2.5.1 Convergence of the path integral

We compute the path integral for the kernel by slicing the clock time into aninfinite number of intervals and integrating over each:

KBA = limN→∞

CN

∫ n=N∏n=1

dtnd~xneıεN+1∑j=1

Lj(2.21)

with CN an appropriate normalization factor.Consider the discrete form of the Lagrangian. We use a tilde to mark the

coordinate time part and an overbar to mark the space part:

Lj ≡ Ltj + L~xj + Lmj (2.22)

Ltj ≡ −am

2

(tj − tj−1

ε

)2

− qatj − tj−1

ε

Φ (xj) + Φ (xj−1)

2(2.23)

L~xj ≡ am

2

(~xj − ~xj−1

ε

)2

+ qa~xj − ~xj−1

ε·~A (xj) + ~A (xj−1)

2(2.24)

Lmj ≡ −abm

2(2.25)

We are using the mid-point rule, averaging the scalar and the vector poten-tials over the start and end points of the step, by analogy with the rule forthree dimensions (Schulman, Grosche and Steiner [76, 35]). For a more generaltreatment, see [44].

Now look at a single step for the free case, vector potential Aµ zero:

∞∫−∞

∞∫−∞

dtjd~xe−ıam (tj−tj−1)2

2ε +ıam(~xj−~xj−1)2

2ε (2.26)

15

The formal tricks normally used to ensure convergence do not work here (e.g.Kashiwa or Zinn-Justin [44, 90]). Perhaps the most popular of these is the useof Wick rotation to shift to a Euclidean time:

τ → −ıτ

This causes integrals to converge rapidly going into the future, but makesthe past inaccessible. For instance, factors of exp (−ıωτ) – which spring upeverywhere in path integrals – converge going into the future, but blow upgoing into the past. If we are to treat time on the same footing as space – ourcentral assumption – then past and future must be treated as symmetrically asleft and right.

Another approach is to add a small convergence factor at a cleverly chosenspot in the arguments of the exponentials. But if we attach a convergencefactor to t and x separately, we break manifest covariance. If we attach ourconvergence factor to both, the fact that the t and x parts enter with oppositesign means any convergence factor that works for one will fail for the other. Wecould try attaching one to the mass m, but this also fails. For instance if a > 0and we subtract a small factor of ıδ from the mass:

m→ m− ıδ (2.27)

the t integral converges but the x integral diverges.We recall the kernel has meaning only when applied to a specific physical

wave function. If we break the incoming wave up into Morlet wavelets and theninto Gaussian test functions, we see that each integral converges by inspection,the factor e−

12 ( t−ls )

2

ensures this. There is no need for convergence factors.3So for physically significant wave functions, there is no problem in the first

place. Effectively we are taking seriously the point that the path integral kernelis a distribution, only meaningful with respect to specific wave functions.

2.5.2 Normalization

We now compute the factor of CN , here working only with the free case. (Weverify the normalization is correct in the general case in appendix on page 78).

Normalization in time We start with the coordinate time dimension only.Consider a Gaussian test function centered on an initial position in coordinatetime t0:

ϕ0 (t0) ≡ 4

√1

πσ2t

e−ıE0t0− (t0−t0)2

2σ2t (2.28)

We write the kernel for the time part as:3A similar effect causes the Feynman loop diagrams to converge, see on page 68.

16

Kτ (tN+1, t0) ∼∫ N∏

j=1

dtje−ıam

N+1∑k=1

(tk−tk−1)2

2ε (2.29)

The wave function after the initial integral over t0 is:

ϕε (t1) =

∫dt0e

−ı am2ε (t1−t0)2

ϕ0 (t0) (2.30)

or:

ϕε (t1) =

√2πε

ıam4

√1

πσ20

√1

f(t)ε

e−ıE0t1+ı

E20

2am ε−1

2σ20f

(t)ε

(t1−t0− E0am ε)

2

(2.31)

with the definition of f (t)τ :

f (t)τ ≡ 1− ı τ

mσ2t

(2.32)

The normalization requirement is:

1 =

∫dt1ϕ

∗ε (t1) ϕε (t1) (2.33)

The first step normalization is correct if we multiply the kernel by a factor of√ıam2πε . Since this normalization factor does not depend on the laboratory time

the overall normalization for N + 1 infinitesimal kernels is the product of N + 1of these factors:

CN ≡√ıam

2πε

N+1

(2.34)

Note also that the normalization does not depend on the specifics of the Gaus-sian test function (the values of E0, σ2

t , and t0) so it is valid for an arbitrarysum of Gaussian test functions as well. And therefore, by Morlet wavelet de-composition, for an arbitrary wave function.

As noted, the phase is arbitrary. If we were working the other way, fromSchrödinger equation to path integral, the phase would be determined by theSchrödinger equation itself. The specific phase choice we are making here hasbeen chosen to ensure the four dimensional Schrödinger equation is manifestlycovariant, see below. We may think of the phase choice as a choice of gauge (seeappendix D on page 79).

Therefore the expression for the free kernel in coordinate time is (with t′′ ≡tN+1, t′ ≡ t0):

Kτ (t′′; t′) =

∫dt1dt2 . . . dtN

√ıam

2πε

N+1

e−ı

N+1∑j=1

( am2ε (tj−tj−1)2)(2.35)

17

Doing the integrals we get:

Kτ (t′′; t′) =

√ıam

2πτe−ıam

(t′′−t′)2

2τ (2.36)

and free wave functions in coordinate time:

ϕτ (t) = 4

√1

πσ2t

√1

f(t)τ

e−ıE0t− 1

2σ2t f

(t)τ

(t−tτ )2+ıE2

02am τ

(2.37)

Normalization in space We redo the analysis for coordinate time for space.We use the correspondences:

t→ x,m→ −m, t0 → x0, E0 → −p0, σ2t → σ2

x (2.38)

With these we can write down the equivalent set of results by inspection. Sincewe will need the results below, we do this explicitly. We get the initial Gaussiantest function:

ϕ0 (x0) = 4

√1

πσ2x

eıp0x0− (x0−x0)2

2σ2x (2.39)

free kernel:

Kτ (x′′;x′) ∼∫dx1dx2 . . . dxNe

ıN+1∑j=1

am2ε (xj−xj−1)2

(2.40)

normalization constant: √2ıπε

am→√

2πε

ıam(2.41)

and kernel:

Kτ (x′′;x′) =

√− ıam

2πτeıam

(x′′−x′)2

2τ (2.42)

The kernel matches the usual (non-relativistic) kernel [25, 76] if a = 1.The wave function is:

ϕτ (x) = 4

√1

πσ2x

√1

f(x)τ

eıp0x− 1

2σ2xf

(x)τ

(x−x0− p0m τ)

2−ı p20

2am τ (2.43)

with the definition of f (x)τ parallel to that for coordinate time (but with

opposite sign for the imaginary part):

f (x)τ = 1 + ı

τ

mσ2x

(2.44)

18

Normalization in time and space The full kernel is the product of thecoordinate time kernel, the three space kernels, and the constant term e−ı

abm2 τ .

We understand x to refer to coordinate time and all three space dimensions:

Kτ (x′′;x′) = −ı a2m2

4π2τ2e−

ıam2τ (x′′−x′)

2−ı abm2 τ (2.45)

We have done the analysis only for Gaussian test functions, therefore byMorlet wavelet decomposition it is completely general.

2.6 Formal expression for the path integralWe now have the path integral:

Kτ (x′′;x′) = limN→∞

∫Dxe

ıN+1∑j=1

Ljε

(2.46)

with the measure:

Dx ≡(−ı m2

4π2a2ε2

)N+1 n=N∏n=1

d4xn (2.47)

and the Lagrangian at each step:

Lj ≡ −am(xj − xj−1)

2

2ε2− aqxj − xj−1

ε

A (xj) +A (xj−1)

2− bm

2(2.48)

Note using the mid-point rule; more sophisticated treatment in [44].

3 Derivation of the Schrödinger equationNormally the path integral expression is derived from the Schrödinger equation.But because for us the path integral provides the defining formulation we needto run the analysis in the “wrong” direction.

Our starting point is a derivation of the path integral from the Schrödingerequation by Schulman [76]. We run his derivation the wrong way and with oneextra dimension.

We start with the discrete form of the path integral. We consider a singlestep of length ε, taking ε→ 0 at the end. Because of this, only terms first orderin ε are needed.

Following Schulman, we define the coordinate difference:

ξ ≡ xj − xj+1 (3.1)

We rewrite the functions of xj as functions of ξ and xj+1. We expand the vectorpotential:

19

Aν (xj) = Aν (xj+1) + (ξµ∂µ)Aν (xj+1) + . . . (3.2)

and the wave function:

ψτ (xj) = ψτ (xj+1) + (ξµ∂µ)ψτ (xj+1) +1

2ξµξν∂µ∂νψτ (xj+1) + . . . (3.3)

giving:

ψτ+ε (xj+1) =√

ıam2πε

√− ıam2πε

3 ∫d4ξe−

ıamξ2

2ε −ıabm2 ε

×eıaqξν(Aν(xj+1)+ 1

2 (ξµ∂µ)Aν(xj+1)+...)

×(ψτ (xj+1) + (ξµ∂µ)ψτ (xj+1) + 1

2ξµξν∂µ∂νψτ (xj+1) + . . .

) (3.4)

We now expand in powers of ξ ∼√ε. We do not need more than the second

power:

ψτ+ε (xj+1) =√

ıam2πε

√− ıam2πε

3 ∫d4ξe−

ıamξ2

2ε

×(

1 + ıaqξνAν (xj+1) + ıaq2 ξνξµ∂µAν (xj+1)− a2q2

2 ξµAµ (xj+1) ξνAν (xj+1)− ıabmε2

)×(ψτ (xj+1) + (ξµ∂µ)ψτ (xj+1) + 1

2ξµξν∂µ∂νψτ (xj+1) + . . .

)(3.5)

The term zeroth order in ξ gives:

√ıam

2πε

√− ıam

2πε

3 ∫dξ4e−

ıamξ2

2ε =

√ıam

2πε

√− ıam

2πε

3√2πε

ıam

√2πε

−ıam

3

= 1 (3.6)

This is not surprising; the normalization above was chosen to do this.Terms linear in ξ give zero when integrated. The terms second order in ξ

(first in ε) are:−ıabmε

2+ ıaq (ξνAν) (ξµ∂µ) +

ıaq

2ξνξµ (∂µAν)

−a2q2

2(ξµAµ) (ξνAν) +

1

2ξµξν∂µ∂ν

ψτ (3.7)

Integrals over off-diagonal powers of order ξ2 give zero. Integrals over diagonalξ2 terms give: √

ıam2πε

∫dξ0e

− ıamξ20

2ε ξ20 = ε

ıam√− ıam2πε

∫dξie

ıamξ2i2ε ξ2

i = − εıam

(3.8)

The expression for the wave function is therefore:

ψτ+ε = ψτ−ıabmε

2ψτ+

qε

m(Aµ∂µ)ψτ+

q

2mε (∂µAµ)ψτ+

ıaq2ε

2mA2ψτ−

ıε

2ma∂2ψτ

(3.9)

20

Taking the limit ε → 0 and multiplying by ı, we get the four dimensionalSchrödinger equation:

ıdψτdτ

= abm

2ψτ +

ıq

m(Aµ∂µ)ψτ +

ıq

2m(∂µAµ)ψτ−

aq2

2mA2ψτ +

1

2ma∂2ψτ (3.10)

or:

ıdψτdτ

(t, ~x) = − 1

2ma

((ı∂µ − aqAµ (t, ~x)) (ı∂µ − aqAµ (t, ~x))− a2bm2

)ψτ (t, ~x)

(3.11)If we make the customary identifications ı ∂∂t → E, −ı~∇ → ~p or ı∂µ → pµ wehave:

ıdψτdτ

= − 1

2ma

((pµ − aqAµ) (pµ − aqAµ)− a2bm2

)ψτ (3.12)

4 Long, slow approximation

4.1 Fixing the scale and additive constantsIn his development of quantum mechanics from a time-dependent perspective[78], Tannor used a requirement of constructive interference in time to derivethe Bohr condition for the allowed atomic orbitals. We use a similar approachhere.

If we average over a sufficiently long period of time, the results will bedominated by the solutions with:

ıdψτ (x)

dτ= 0 (4.1)

The argument here is not that the variation from the long, slow approx-imation solution is small, but that over time interactions with the system inquestion will tend to be dominated by those that do match. Decoherence withthe environment will tend to reinforce the stationary solutions. And as will“internal decoherence”, the self interactions of the wave function[15].

Accepting this, then the right side looks like the Klein-Gordon equation. Tocomplete this identification, first look at the case with the vector potential Azero: (

p2 − a2bm2)ψ = 0→ a2b = 1 (4.2)

Now when A is not zero we have:((p− aqA)

2 −m2)ψ = 0→ a = 1→ b = 1 (4.3)

We will refer to this as the “long, slow approximation”.

21

In the free case, the long, slow approximation picks out the on-shell compo-nents: (

p2 −m2)ψ = 0 (4.4)

And more generally the solutions of the Klein-Gordon equation with the minimalsubstitution p→ p− qA: (

(p− qA)2 −m2

)ψ = 0 (4.5)

The two constants are now fixed. For the record, the full Schrödinger equa-tion is:

ıdψτdτ

(t, ~x) = − 1

2m

((ı∂µ − qAµ (t, ~x)) (ı∂µ − qAµ (t, ~x))−m2

)ψτ (t, ~x) (4.6)

and in momentum space:

ıdψτdτ

= − 1

2m

((pµ − qAµ) (pµ − qAµ)−m2

)ψτ (4.7)

And the free Schrödinger equation is:

2mıdψτdτ

(t, ~x) =(∂µ∂

µ +m2)ψτ (t, ~x) (4.8)

and in momentum space:

2mıdψτdτ

= −(pµp

µ −m2)ψτ (t, ~x) (4.9)

4.2 Physical meaning of the long, slow approximationTo see the relevant scale, we estimate the clock frequency f :

f ∼ −E2 − ~p2 −m2

2m(4.10)

We will argue (below on page 25) that in the non-relativistic case E is oforder mass plus kinetic energy:

E ∼ m+~p2

2m(4.11)

So we have:

E2 − ~p2 −m2 ∼(m+

~p2

2m

)2

− ~p2 −m2 =

(~p2

2m

)2

(4.12)

This is the just the kinetic energy of the atom, squared. We have:

~p2

2m∼ eV (4.13)

22

But the denominator is of order MeV . So we can estimate the clock fre-quency f as:

f ∼ eV 2

MeV∼ eV · 10−6 (4.14)

Energies of millionths of an electron volt 10−6eV correspond to times of or-der millions of attoseconds 106as or picoseconds, a million times longer thanthe natural time scale of the effects we are looking at. So the long, slow approx-imation seems reasonable. With that said, we will focus on effects that do notdepend on the rate. As we derived it by extending the classical equations intothe quantum realm, it can only at best give us a reasonable first approximation.

4.3 Can we make an observer independent choice of frame?We have one further concern before applying this. Whose clock should we useas the laboratory clock?

If we did the above derivation for Bob rather than Alice, we would see hisclock time τ → γτ where γ ≡ 1

/√1− v2 where v is his velocity relative to hers.

We could argue that if Bob is not going that quickly, that the errors createdby this will introduce only small corrections, which when combined with thesmall corrections we expect, will be of second order and therefore not relevantfor falsifiability.

However establishing frame independence is interesting as a point of princi-ple. This may be done in a natural way by making use of an observation fromWeinberg [85]:

We may treat the Einstein field equation for general relativity as representingconservation of energy-momentum when exchanges of energy momentum withspacetime are included.

Consider the Einstein field equations:

Gµν ≡ Rµν −1

2gµνR = −8πGTµν (4.15)

Rewrite as:

(Gµν + 8πGTµν);ν = 0 (4.16)

We may therefore associate an energy momentum tensor (tµν in Weinberg’snotation) with local space time. Define:

gµν = ηµν + hµν (4.17)

where hµν vanishes at infinity but is not assumed small. The part of the Riccitensor linear in h is:

R(1)µν ≡

1

2

(∂2hλλ∂xµ∂xν

−∂2hλµ∂xλ∂xκ

− ∂2hλκ∂xλ∂xµ

+∂2hµν∂xλ∂xλ

)(4.18)

23

The exact Einstein equations may be written as:

R(1)µν −

1

2ηµκR

(1)λλ = −8πG (Tµν − tµν) (4.19)

where tµν is defined by:

tµν ≡1

8πG

(Rµν −

1

2gµκR

λλ −R(1)

µν +1

2ηµκR

(1)λλ

)(4.20)

We may interpret tµν as the energy-momentum of the gravitational field itself.We change to a coordinate frame in which tµν is diagonalized. We will refer

to this as the rest frame of the vacuum. We can treat this as the defining framefor the four dimensional Schrödinger equation. As this frame is invariant (up torotations in three-space) we have an invariant definition of the four dimensionalSchrödinger equation. The results for Alice or Bob may be derived from this byappropriate Lorentz transformations.

5 Free solutionsWe now look at the birth, life, and death of a free particle.We look in turn atthe initial wave function, the evolution of the wave function as a function ofclock time, and its detection at a specific instant. For the most part, we focuson the case of a single space dimension.

5.1 Physically reasonable definition of the initial wave func-tion

To compute the initial wave function we will look for the solutions for particlesin a constant potential. The free case is of course the case with constant po-tential zero; the more general case is no more work to do here and useful whenestimating the bound state wave functions.

We will compute this, first using separation of variables and then using amaximum entropy approach. The solution by separation of variables is simpler;the solution by maximum entropy more physically plausible.

5.1.1 Solution by separation of variables

We consider the Klein-Gordon equation for potentials constant in time. Weassume the magnetic field is zero:((

E − V (~x))2 − ~p2 −m2

)ϕ (t, ~x) = 0 (5.1)

This includes attractive potentials as well as scattering potentials, provided theyare constant in time. Expanded, the equation is:(

− ∂2

∂t2− 2V (~x) ı

∂

∂t+ V (~x)

2 − ~p2 −m2

)ϕ (t, ~x) = 0 (5.2)

24

We solve this using separation of variables, looking for solutions of the form:

ϕ (t, ~x) = φn (t) ϕn (~x) (5.3)

We assume we have a solution for the standard Klein-Gordon equation:((En − V (~x)

)2 − ~p2 −m2)ϕn (~x) = 0 (5.4)

Then the coordinate time part is:

φn (t) =1√2πe−ıEnt (5.5)

Because the potential is constant in time each use of the operator E turns intoa constant En via:

E → ı∂

∂t→ En (5.6)

We get immediately:((En − V (~x)

)2 − ~p2 −m2)ϕn (t, ~x) = 0 (5.7)

So every solution of the Klein-Gordon equation in SQM generates a corre-sponding solution in TQ.

However this is not entirely satisfactory. We have a solution which is “fuzzy”in space, but “crisp” in time. This is implausible.

We would expect the dispersion in energy would be comparable to the dis-persion in momentum of the space part:

σ2E ∼ σ2

px + σ2py + σ2

pz (5.8)

A more realistic, if more complex solution, would include off-shell compo-nents. While mathematically acceptable, our solution is not physically plausible.

5.1.2 Solution by maximum entropy

The long, slow approximation picks out a single solution. But in practice weexpect there would be a great number of solutions, with the one given by thelong slow approximation merely the most typical.

To get a more physically reasonable solution we look for a probability densityfunction which satisfies the relevant constraints.

As many probability density functions will do this, we will use the methodof Lagrange multipliers to select the one with maximum entropy.

From the probability density, we will infer the wave function4.We start with the free case, as representing the simplest case of a constant

potential. Do bound case below. To help prepare for that, we assume are given4As often in this analysis, we are going “the wrong way”, here from probability distribution

to wave function.

25

the wave function in three dimensions φ (~x) and will shift back to one spacedimension in the subsection.

We take as the constraints the norm, the expectation of the energy, and theexpectation of the energy squared:

〈1〉 = 1

E ≡ 〈E〉 =

√m2 + 〈~p〉2⟨

E2⟩

=⟨m2 + ~p2

⟩= m2 +

⟨~p2⟩ (5.9)

Uncertainty defined as:

∆E ≡√〈E2〉 − 〈E〉2 (5.10)

The expectations are defined as integrals over the probability density

〈f〉 ≡∫d~pρ (~p) f (~p) (5.11)

The constraints imply ∆E = ∆p.We will work with box normalized energy eigenfunctions:

φn (t) ≡ 1√2T

e−ıE0nt (5.12)

E0 ≡π

T(5.13)

where n is an integer running from negative infinity to positive and the eigen-functions are confined to a box extending T seconds into the future and Tseconds into the past, where T is much larger than any time of interest to us.

A general wave function can be written as:

ψ (t) =

∞∑n=−∞

cnφn (t) (5.14)

The coefficients c only appear as the square:

ρn ≡ c∗ncn (5.15)

Expressed in this language we have the constraints:

C0 ≡∞∑

n=−∞ρn − 1 = 0

C1 ≡ E0

∞∑n=−∞

nρn − 〈E〉 = 0

C2 ≡ E20

∞∑n=−∞

n2ρn −⟨E2⟩

= 0

(5.16)

We would like to find the solution that maximizes the entropy:

26

S ≡∞∑

n=−∞−ρn ln (ρn) (5.17)

We form the Lagrangian from the sum of the entropy and the constraints, withLagrange multipliers λ0, λ1, and λ2:

L ≡ S + λ0C0 + λ1C1 + λ2C2 (5.18)

To locate the configuration of maximum entropy, we take the derivative withrespect to the ρn:

∂L

∂ρn= 0 (5.19)

getting:

− ln ρn − 1 + λ0 + E0nλ1 + E20n

2λ2 = 0 (5.20)

Therefore the distribution of the ρn is given by an exponential with zeroth, first,and second powers of the energy:

ρ (E) ∼ e−a−bE−cE2

(5.21)

The constraints force the constants:

ρ (E) =1√

2π∆E2e−

(E−E)2

2∆E2 (5.22)

We therefore have the probability density in energy; now we wish to estimatethe wave function in energy (or equivalently time). There are two distinct waysto do this. First for each value of the wave function in momentum, we can positan associated delta function in energy:

ϕ (E, p) ∼ δ (E − Ep) ˆϕ (p) (5.23)

Ep ≡√m2 + p2 (5.24)

Or we can write the total wave function as the direct product of a wave functionin energy times the postulated wave function in momentum:

ϕ (E, p) ∼ ˆϕ (E) ˆϕ (p) (5.25)

The first approach implies a detailed specification of the distribution of thewave function in energy. If the wave function in momentum has a complex shape,so too will the energy part. This seems a bit overconfident for a first attack onthe problem. We therefore adopt the second approach as more appropriate toan initial understanding.

We start with a Gaussian test function in momentum:

27

ˆϕ0 (p) = 4

√1

πσ2p

e−ı(p−p0)x0− (p−p0)2

2σ2p (5.26)

With a corresponding wave function in space:

ϕ0 (x) = 4

√1

πσ2x

eıp0x− (x−x0)2

2σ2x (5.27)

The wave function in momentum implies by the arguments above a wave func-tion of the form:

ˆϕ0 (E) = 4

√1

πσ2E

eı(E−E0)t0− (E−E0)2

2σ2E (5.28)

σ2E = σ2

p (5.29)

E0 =√m2 + p2 (5.30)

Taking the Fourier transform of the energy part we have:

ϕ0 (t) = 4

√1

πσ2t

e−ıE0(t−t0)− t2

2σ2t (5.31)

σ2t =

1

σ2E

(5.32)

We set t0 = 0 as the overall phase is already supplied by the space/momentumpart.

The full wave functions are the products of the coordinate time and space(or energy and momentum) parts:

ϕ0 (t, x) = ϕ0 (t) ϕ0 (x) (5.33)

ϕ0 (E, p) = ˆϕ0 (E) ˆϕ0 (p) (5.34)

5.1.3 Bound state wave functions

We use this approach to estimate the dispersion of a bound wave function intime.

In the case of a Coulomb potential we can estimate ∆p from the virialtheorem: ⟨

~p2

2m

⟩= −1

2〈V 〉 (5.35)

which implies:

28

⟨~p2

2m

⟩+ 〈V 〉 = En →

⟨~p2

2m

⟩= −En (5.36)

Since the average momentum is zero, we have the estimate:

∆E =√−2mEn (5.37)

Substituting the mass of the electron and the Rydberg constant, we have:

∆E =√

2 · 13.6eV · (0.511 · 106) eV = 3728 eV (5.38)

And the corresponding dispersion in coordinate time is:

∆t =~

∆E=

6.582 · 10−16eV · sec

3728eV= .1766as (5.39)

This is essentially the same as we started with, though using the energy ratherthan the radius to estimate the width in coordinate time.

We write as a direct product.

ϕ(nlm) (t) ψ(nlm) (~r)

with:

ϕ(nlm) (t) = ϕ

(σt =

1

σnlm

)This should be good enough for order of magnitude estimates and therefore

falsifiability.

5.2 Evolution of the free wave function in laboratory timeWe use the non-relativistic approximation on the SQM side. It seems in generaleasier to compare TQ to the non-relativistic SQM, even though a comparisonto the solutions of the Klein-Gordon equation might seem “fairer”.

5.2.1 Evolution in SQM

The non-relativistic Schrödinger equation is [57]:

ı∂

∂τψ =

p2

2mψ = − ∂2

x

2mψ (5.40)

At clock time zero we start with a Gaussian test function with position x0,average momentum p0, and dispersion σp:

ˆϕ0 (p) = 4

√1

πσ2p

e−ıpx0− (p−p0)2

2σ2p (5.41)

In momentum space the problem is trivial. The solution is:

29

ˆϕτ (p) = 4

√1

πσ2p

e−ıpx0− (p−p0)2

2σ2p−ı p

2

2m τ (5.42)

In coordinate space we get:

ϕτ (x) = 4

√1

πσ2x

√1

f(x)τ

eıp0x− 1

2σ2xf

(x)τ

(x−x0− p0m τ)

2−ı p20

2m τ (5.43)

with:

σx =1

σp(5.44)

f (x)τ = 1 + ı

τ

mσ2x

(5.45)

5.2.2 Evolution in TQ

In two dimensions the Schrödinger equation for TQ is:

ıdψτdτ

(t, ~x) = −E2 − p2 −m2

2mψτ (t, ~x) =

(∂2t

2m− ∂2

x

2m+m

2

)ψτ (t, ~x) (5.46)

This looks very much like the NR, but with one extra space dimension.We start in energy momentum space. The momentum part is as above. The

wave function in energy, with average time at start t0, energy E0, and dispersionσE , is:

ˆϕ0 (E) ≡ 4

√1

πσ2E

eıEt0− (E−E0)2

2σ2E (5.47)

Again we merely push this forward in clock time:

ψτ (E, p) = ˆϕ0 (E) ˆϕ0 (p) exp

(−ıE

2 − p2 −m2

2mτ

)(5.48)

We divide up the pieces of the clock time part, assigning the E2

2m to theenergy part, the p2

2m to the momentum part, and keeping the third part outside:

ψτ (E, p) = ˆϕτ (E) ˆϕτ (p) exp(ım

2τ)

(5.49)

Now the energy part works in parallel to the momentum part:

ˆϕτ (E) ≡ 4

√1

πσ2E

eıEt0− (E−E0)2

2σ2E

−ıE20

2m τ (5.50)

And in coordinate space:

30

ϕτ (t) = 4

√1

πσ2t

√1

f(t)τ

e−ıE0t+ı

E20

2m τ−1

2σ2t f

(t)τ

(t−t0−E0m τ)

2

(5.51)

withσE =

1

σt(5.52)

f (t)τ ≡ 1− ı τ

mσ2t

(5.53)

with the expectation for coordinate time

tτ = t0 +E0

mτ (5.54)

implying a velocity for coordinate time with respect to laboratory time

v0 =E0

m(5.55)

We define as usual:

γ ≡ E

m(5.56)

And have in the non-relativistic case, γ ≈ 1. So the average of the coordinateadvances at the traditional one-second-per-second rate relative to the clock time.

5.2.3 Comparison of TQ to SQM

There is a problem, however. In the long, slow approximation we have:

E2 − p2 −m2

2m≈ 0 (5.57)

So in SQM we have a wave function oscillating as p2

2m while in TQ we seem tohave one that doesn’t oscillate at all. Since oscillation in clock time will generateinterference patterns – as in double-slit in time experiment – this does not seemto make sense.

We need to work this out explicitly. The key point is that the dependenceon clock time is primarily mediated via the dependence on coordinate time

The total derivative with respect to the clock time is given by:

ıd

dτ= ı

∂

∂τ+ ı

dt

dτ

∂

∂t(5.58)

The partial derivative with respect to clock time is expected, in the long,slow approximation, to be about zero (as above on page 22):

ı∂

∂τ≈ 0 (5.59)

31

In the non-relativistic case, the derivative of t with respect to clock time isabout one:

dt

dτ≈ 1 (5.60)

To check this, look at the partial derivative with respect to coordinate time,which defines the energy operator:

ı∂

∂t= E

The expectation of the energy operator gives the average frequency:⟨E⟩

= E

In the non-relativistic case this is approximately give by:

E ≈ m+p2

2m

We may discard the m part as constant and therefore irrelevant:

m+p2

2m→ p2

2m

so the total derivative is about the same as in the SQM.As a side effect we have that the expectation of the coordinate time is about

equal to the clock time. While Alice’s dog is always getting ahead of and behindher, on average his position is about equal to hers:

〈t〉 ≈ τ (5.61)

at least in the non-relativistic case. We will have to wait till we treat themultiple particle case (below on page 50) before we can consider the problem ofrelativistic dogs.

5.3 Time of arrival measurementsTo investigate the question of measurement we look specifically at the mea-surement of time of arrival. We assume we have a particle going left to right,starting at x = 0. We place a detector at position x = L. It records when itdetects the particle. The metric we are primarily interested in is the dispersionin time of arrival at the detector.

We will assume that if the detector fires at clock time τ then the coordinatetime of the particle that caused it to fire is to be taken as t = τ . This is requiredby our principle of maximum symmetry between time and space. If a detectorlocated at position X fired when hit by a particle we would take the particle’sposition in space as x = X. So by the same principle, detection by a detectorat clock time must imply t = τ .

32

Figure 5.1: Time of arrival in SQM

By the same token if a particle is emitted by a source at position X wewould take the start position of the path as x = X. And therefore if a particleis emitted by a source that is only turned on for a brief instant at clock time Twe must take the start position of the path as t = T .

And if we know both source and detector positions in space and time than allcorresponding paths are clamped at both ends. In between source and detectorthe paths can examine all sorts of interesting spaces but each path is clampedat the endpoints. Alice and her dog leave from the same starting point in spacetime and arrive at the same ending point in space time, but while classical Alicetakes the shortest path between the start and end points, the quantum dogexplores all paths.

We have one further problem before we can get to our analysis: paths in TQare much more complex than those in SQM. If a detector is a camera shutter,open for a fraction of a second, then any paths that arrive early or late willmerely be “eaten” by the closed shutter. But what if our apparatus can somehow

33

be toggled from transparent to absorptive and back, as via “electromagneticallyinduced transparency”[29]? Then the paths can arrive early but then circleback, or arrive late but circle forward, or even perform a kind of drunkard’swalk around the detector till they choose to fall into it.

Since we are primarily interested in comparisons of TQ to SQM, rather thanin fully exploring the elaborations of TQ, we will focus (as it were) on the camerashutter model. Paths that arrive early or late will be thrown on the floor or fallout in the averaging. We will pay careful attention to the normalization, butnot otherwise worry about sobriety-challenged paths.

5.3.1 Metrics

With that settled, to compute the dispersion in time, we log how many hits weget in each time interval:

ρ (τ) (5.62)

then calculate the average:

〈τ〉 ≡∞∫−∞

dτρ (τ) (5.63)

and the uncertainty:

〈∆τ〉2 ≡∞∫−∞

dττ2ρ (τ)− 〈τ〉2 (5.64)

5.3.2 Time of arrival in SQM

x

SQM

Source

Detector

TQ

LD

SQMTQ

CM

Figure 5.2: Time of arrival in TQ

34

We start with a particle with initial position x = 0 and with average mo-mentum (in the x direction) of p0. We will assume that the initial dispersion inmomentum is small. We have the wave function from above. The probabilitydensity is then:

ρτ (x) ≡ |ϕτ (x)|2 =

√√√√ 1

πσ2x

∣∣∣f (x)τ

∣∣∣2 e− (x− p0

mτ)

2

σ2x|f(x)

τ |2 (5.65)

To get the dispersion in time, we integrate over possible detection times.We expand around the time the wave function is most likely to cross the

plane of the detector:

τD ≡mLDp0

=LDv

(5.66)

At τ we can write x as x = x+ δx. We can turn this around, and write:

τD = τD + δτD (5.67)

Or to lowest order in δτ :

δx = −vdδτ (5.68)

We expand the numerator in the exponential in powers of δτ :

ρτ (L) ≈√√√√ 1

πσ2x

∣∣∣f (x)τ

∣∣∣2 e− (vδτ)2

σ2x|f(x)

τ |2 (5.69)

And the denominator to second order in powers of δτ :

σ2x

∣∣∣f (x)τ

∣∣∣2 = σ2x +

τ2

m2σ2x

= σ2x +

(τ + δτ)2

m2σ2x

≈ (τ + δτ)2

m2σ2x

≈ τ2

m2σ2x

(5.70)

Giving:

ρδτ ≈√v2m2σ2

x

πτ2e−

v2m2σ2x

τ2 (δτ)2

(5.71)

We define an effective dispersion in time:

στ ≡1

mvσxτ (5.72)

And the probability of detection as:

ρδτ =

√1

πσ2τ

e− (δτ)2

σ2τ (5.73)

35

This is correctly normalized to one, centered on τ = τ , and with uncertainty:

∆τ =1√2στ (5.74)

Particularly important is the inverse dependence on the velocity. Intuitivelyif we have a slow moving (non-relativistic) particle, it will take a long time topass through the plane of the detector, causing the associated uncertainty intime to be relatively large.

Comparison to a time-of-arrival operator As a cross-check, we compareour treatment to the time of arrival operator analysis in Muga and Leavens [63](following Kijowski [47]). They give a probability density in time of:

ρ (τ) =

∣∣∣∣∣∣∞∫

0

dp

√p

me−ı

p2τ2m ˆϕ (p) exp (ıpL)

∣∣∣∣∣∣2

+

∣∣∣∣∣∣0∫

−∞

dp

√−pme−ı

p2τ2m ˆϕ (p) exp (−ıpL)

∣∣∣∣∣∣2

(5.75)where ˆϕ is an arbitrary momentum space wave function normalized to one.Assume:

p0 � σp (5.76)

so:

p ≈ p0 (5.77)

If our wave functions are closely centered on p, the term with negative mo-mentum can be dropped or even flipped in sign without effect on the value ofthe integral. Further, by comparison to the exponential part, the term underthe square root is roughly constant:√

p

m≈√p0

m(5.78)

So we may replace Muga and Leaven’s expression by the simpler:

ρ (τ) ≈ p0

m

∣∣∣∣∣∣∞∫−∞

dpe−ıp2τ2m ˆϕ (p) exp (ıpL)

∣∣∣∣∣∣2

(5.79)

As the contents of the integral are the Fourier transform of the (clock) timedependent momentum space wave function, it is the clock time dependent spacewave function at x = L:

ρ (τ) ≈ p0

m|ϕτ (L)|2 (5.80)

And the rest of the analysis proceeds as above.

36

5.3.3 Time of arrival with TQ

It is striking that there is considerable uncertainty in time even when time istreated classically. Our hypothesized uncertainty in time will be added to thispre-existing uncertainty.

Using the time wave function from above we have for the probability densityin time:

ρτ (t) =

√√√√ 1

πσ2t

∣∣∣f (t)τ

∣∣∣2 e− 1

σ2t |f(t)

τ |2(t−τ)2

(5.81)

We multiply by the space part from above to get the full probability density:

ρD (t, LD) = ρD (t) ρD (LD) (5.82)

If t were replaced by space dimension y, we would have no doubt as to how toproceed. To get the overall uncertainty in y we would integrate over clock time

(∆y)2

=

∫dy (y − y)

2∫dτρτ (y) ρτ (LD) (5.83)

Therefore we write (taking y → t):

(∆t)2

=

∫dt (t− τ)

2∫dτ ρτ (t) ρτ (LD) (5.84)

This is a convolution of clock time with coordinate time. To solve we first invokethe same approximations as above:

σ2t

∣∣∣f (t)τ

∣∣∣2 = σ2t + τ2

m2σ2t

= σ2t + (τ+δτ)2

m2σ2t≈ (τ+δτ)2

m2σ2t≈ τ2

m2σ2t

στ ≡ τmσt

ρτ (t) ≈√

1πσ2

τe− 1σ2τ

(t−τ)2

(5.85)

So we have for the full probability distribution in t:

ρτ (t) ≡∫dτ√

1πσ2

τe− 1σ2τ

(t−τ)2√1πσ2

τe− 1σ2τ

(τ−τ)2

(∆t)2

=∫dt (t− τ)

2ρτ (t)

(5.86)

The convolution over τ is trivial giving:

ρτ (t) =

√1

πσ2τ

e− (t−τ)2

σ2τ

We therefore have for the total uncertainty at the detector:

σ2τ = σ2

τ + σ2τ

37

So the dispersion in time is the sum of the dispersions from the time andthe space parts. This is intuitively reasonable. Restating the definitions for thetwo dispersions:

σ2τ =

τ2

m2v2σ2x

σ2t ≈

τ2

m2σ2t

(5.87)

From the long, slow approximation, we would expect particle wave functionsto have initial uncertainties in energy/time comparable to their uncertaintiesin momentum/space. σt ∼ σx. But the conventional contribution has an addi-tional 1

v in it. Since in the non-relativistic case, v � 1 the total uncertainty willbe dominated by the space part.

This motivates looking at the relativistic and therefore necessarily the mul-tiple particle case (below on page 50).

6 Semi-classical approximation

6.1 Derivation of the semi-classical approximationIn the path integral formulation of quantum mechanics, the semi-classical in-terpretation provides a natural bridge between the classical and the quantumpictures. Essentially it is the saddle point/steepest descents approximation ap-plied to path integrals.

To locate the saddle point we rewrite the coordinates in the Lagrangian inon page 19 as xj = xj + δxj . To get the saddle point, we define x by therequirement that the first variation with respect to x be zero:

0 = −m(xµj−x

µj−1

ε2

)+m

(xµj+1−x

µj

ε2

)− q

2ε (Aµ (xj−1)−Aµ (xj+1))− q2ε

(xνj+1 − xνj−1

) ∂Aν(xj)

∂xjµ

(6.1)

In the continuum limit this is the classical equations of motion (appendixon page 77):

mxµj = −qdAµjdτ

+qdxνjdτ

∂Aν (xj)

∂xjµ= −q

(∂Aµ (xj)

∂xjν− ∂Aν (xj)

∂xjµ

)dxνjdτ

= qFµν (xj) xνj

(6.2)We therefore identify x as the classical path. The full expression for the

kernel is then:

Kτ (x′′;x′) =

∫Dx exp

ıε j=N+1∑j=1

(Lj +

1

2

∂2Lj∂δx∂δx

δxδx+O (δx)3

) (6.3)

38

with:

Lj ≡ −m

2

(xj − xj−1

ε

)2

− e xj − xj−1

ε

(A (xj) +A (xj−1)

2

)− m

2ε (6.4)

This is exact when we include the cubic and higher terms. Including onlyterms through the quadratic we get:

Kτ (x′′;x′) ≈ Fτ (x′′;x′) Kτ (x′′;x′) (6.5)

with:

Kτ (x′′;x′) = exp

ıε j=N+1∑j=1

Lj

→ exp(ıS(classical)τ

)(6.6)

and fluctuation factor F :

Fτ (x′′;x′) ≡∫Dx exp

ıε j=N+1∑j=1

1

2

∂2Lj∂δxµ∂δxν

δxµδxν

(6.7)

The fluctuation factor can be computed in terms of the action and its deriva-tives. The derivation is nontrivial. It is done in the one dimensional case in [73]and in one dimension and higher dimensions in [48]. Fortunately these deriva-tions are independent of the number of space dimensions. We up this fromthree to four, treating coordinate time as a fourth space dimension x4. Withderivation complete we replace x4 with ıt.

Fτ (x′′;x′) =1√

2π4

√det

(−∂

2Sτ (x′′;x′)

∂x′∂x′′

)(6.8)

With final result:

Kτ (x′′;x′) ≈ 1√

2π4

√det

(−∂

2Sτ (x′′;x′)

∂x′∂x′′

)exp (ıSτ (x′′;x′)) (6.9)

The semi-classical approximation is exact when there are no third orderterms, true for the free, constant magnetic field, and constant electric fieldcases, which we now treat in turn.

6.2 Derivation of the free kernel using the semi-classicalapproximation

The semi-classical approximation is arguably the simplest way to derive the freepropagator. Since the free Lagrangian has no terms higher than quadratic, thesemi-classical approximation is exact.

39

Lagrangian:

L (x, x) = −m2x2 − m

2(6.10)

Euler-Lagrange equation:

mx = 0 (6.11)

Classical path:

xτ ′ =∆x

ττ ′ + x (6.12)

The Lagrangian along the classical path is a constant:

L = −m2

(∆x

τ

)2

− m

2(6.13)

The action:

S = −m2τ

(∆x)2 − m

2τ (6.14)

Determinant of the action:

det

(−∂

2S (x′′;x′)

∂x′′∂x′

)= −m

4

τ4(6.15)

And the full kernel:

Kτ (x′′;x′) = − m2

4π2τ2exp

(− ım

2τ(x′′ − x′)2 − ım

2τ)

(6.16)

A table of the free kernels is assembled as an appendix E on page 80.

6.3 Constant magnetic fieldOne of the strengths of TQ is a more complete symmetry between treatmentsof magnetic and electric cases.

We look at the case of a constant magnetic field in the z direction ~B =(0, 0, B) with vector potential ~A = (0, Bx, 0). We ignore the mass term, whichcan be gauged away . The Lagrangian is then:

L = −m2t2 +

m

2~x

2+q

2Byx (6.17)

The kernel splits up into t and z pieces, which are just the free kernels, andan x, y piece, which is the standard magnetic kernel:

Kτ (t′, x′, y′, z′; t, x, y, z) = K(free)τ (t′; t) K(free)

τ (z′; z) K(mag)τ (x′, y′;x, y)

(6.18)We define the Larmor frequency:

40

ω ≡ qB

m(6.19)

Kleinert [48] takes advantage of the formal resemblance of the magnetic partto the harmonic oscillator to compute the classical action:

S(mag)τ (x′, y′;x, y) =

m

2

(ω2

cot(ωτ

2

)((∆x)

2+ (∆y)

2)

+ ω (xy′ − x′y))(6.20)

Giving fluctuation factor (in two dimensions):

det

(−∂′∂S

(mag)τ

∂x′∂x

)= − ı

4π

mω

sin(ωτ2

) (6.21)

And full kernel:

K(mag)τ (x′, y′;x, y) = − ı

4π

mω

sin(ωτ2

) exp(ıS(mag)τ (x′, y′;x, y)

)(6.22)

As a quick check, we note we recover the free form in the limit ω → 0.Note that the TQ and SQM kernels are fully equivalence. The changes in

TQ relate to the time part, which is factored out.

6.4 Constant electric fieldNext we look at the case of a constant electric field in the x direction ~E =(E, 0, 0) which will be produced by a potential Φ = −Ex or in four dimensionalnotation A = (−Ex, 0, 0, 0). The Lagrangian is now:

L = −mt2

2+m

2~x2 +

qE

2tx (6.23)

The kernel factors as before:

Kτ (t′, x′, y′, z′; t, x, y, z) = K(free)τ (y′, z′; y, z)K(elec)

τ (t′, x′; t, x) (6.24)

We define, analogously to the Larmor frequency:

α ≡ qE

m(6.25)

The integrals are formally equivalent to the magnetic case with the substi-tutions:

y → −ıtt→ ıyω → ıα

(6.26)

41

This will be recognized as a variation on the duality of the electric andmagnetic fields, which in turn comes from the ability to rotate space and timeinto each other in special relativity.

Therefore we can write the action, fluctuation factor, and kernel by inspec-tion:

S(elec)τ (t′, x′; t′, x′) =

m

2

(α2

coth(ατ

2

)(− (∆t)

2+ (∆x)

2)

+ α (xt′ − x′t))

(6.27)Again we note the fluctuation factor for the electric part comes from the

determinant of a two by two matrix:

det

(−∂′′∂′S

(elec)τ ′′τ ′

∂x′′∂x′

)=m2α2

4

(− coth2

(α∆τ

2

)+ 1

)= − m2α2

4 sinh2(α∆τ

2

)(6.28)

K(elec)τ ′′τ ′ (t′′, x′′; t′, x′) =

mα

4π sinh(α∆τ

2

) exp(ıS

(elec)τ ′′τ ′ (t′′, x′′; t′, x′)

)(6.29)

The limit as α→ 0 should give the free kernel. Again, easily verified.

6.4.1 Failure of Duality in SQM

Note that the kernels for TQ and SQM are not equivalent. If we start with thesame Lagrangian in SQM, we will only include variations in x, not t, so thatthe fluctuation factor will only depend on one dimension, not two.

The situation is analogous to the free case in that in TQ, quantum fluc-tuations in the time direction are included by definition, but in SQM suchfluctuations are ruled out – also by definition. Temporal quantization is – withrespect to time – as SQM is to classical mechanics with respect to the spacedimensions.

But if in SQM the fluctuation factor for the magnetic field depends on twodimensions but the kernel for the electric field depends on only one, then thetwo kernels violate duality. In SQM the kernels for the magnetic and electrickernels cannot be equivalent under exchange of the electric and magnetic fields.

This is not a surprise. The duality of the electric and magnetic fields followsfrom special relativity. TQ by construction respects this fully; SQM does not.Given that the electric fields are the time-space components of the Fµν tensor,while the magnetic fields correspond to the space-space components of F , it isclear that given the special handling of time in SQM transformations betweenthe two may be problematic.

This in turn means that duality provides another way to produce experimen-tal tests of TQ. For instance, we could look at variations on the Aharonov-Bohmeffect, especially those involving the time-dependent electric fields:[5, 83, 84].

42

7 Single and double slit experimentsThe investigation of diffraction in time started over sixty years ago with Moshin-sky [60, 61] and continues, with a recent review by Gerhard and Paulus [30].Particularly interesting for our purposes are treatments in Umul [81] and in apair of papers by Marchewka and Schuss [55, 56], which look specifically at thescattering of wave packets. Even in SQM, a full treatment of scattering by atime gate is a complex problem. And here we further require an approach whichwill make a comparison of the results with and without TQ as straightforwardas possible.

We will ignore paths that loop back and forth through the gate. Mostelementary treatments make this assumption of a single passage (for an analysisof the effects of multiple passages of a gate see for Yabuki, Raedt, and Sawant[88,22, 74]).

We will also, following Feynman and Hibbs [25], take the gate as having aGaussian shape, rather then turning on and off instantly. This results in thewave functions at the detector also being Gaussian in shape. With a sharpedged gate, the wave functions at the detector become sums of error functions,and significantly more complex.

Note there is a close relation between a single slit and measurement. Ameasurement may be thought of as a single slit where the paths cross the slitonly once and are then harvested with perfect efficiency.

7.1 Single slit in time in standard quantum mechanics

Source Gate Detector

W

Clipped

Unclipped

Unclipped

Clipped

Figure 7.1: Single slit in time in standard quantum mechanics

We start with a wave function in x:

43

ϕτ (x) = 4

√1

πσ2x

√1

f(x)τ

eıp0x− 1

2σ2xf

(x)τ

(x−x0− p0m τ)

2−ı p20

2m τ (7.1)

or in p:

ˆϕτ (p) = 4

√1

πσ2p

e−ıpx0− (p−p0)2

2σ2p−ı p

2

2m τ (7.2)

We take the velocity v ≡ p0

m . We again assume the particle is non-relativisticso that v � 1.

The gate is located at x = B, centered on clock time A, with width in timeW :

Gτ = e−(τ−A)2

2W2 (7.3)

For simplicity we center the particle beam on the gate. If τG is the averagetime at which the particle reaches the gate, we arrange it so that:

τG = A (7.4)

This gives:

v =B

A(7.5)

With the detector at position x = L, we define T as the average time atwhich the particle is detected:

T ≡ vL (7.6)

The evolution of the wave function is simplest in the p basis, but the gatedefined in τbasis. To switch back and forth between these two points of viewwe define:

p = p0 + δp = mv + δp

τ = τ + δτ(7.7)

We assume that the incoming wave function can be treated as a sum of prays. This worked in the time of arrival case, has the merit of simplicity, andlets us make a direct comparison between TQ and SQM. And this lets us changethe basis from p→ τ and back using:

p = mx

τ(7.8)

We assume the beam is reasonably well-focused and use a para-axial approx-imation, to quadratic order:

44

δp = mx

τ + δτ−mx

τ≈ mx

τ

(−δττ

+δτ2

τ2

)δτ = m

x

p0 + δp−m x

p0≈ x

v

(−δpp0

+δp2

p20

) (7.9)

At the gate, the wave function in p space is:

ˆϕG(−) (p) = 4

√1

πσ2p

e− (δp)2

2σ2p−ı p

2

2m τG (7.10)

In τ space:

ϕG(−) (δτ) = 4

√1

πσ2G

e− δτ2

2σ2G

−ı 12m ( mB

A+δτ )2(A+δτ)

= 4

√1

πσ2G

e− δτ2

2σ2G

−ımB2

2A

(1− δτA + δτ2

A2

)

(7.11)with σG defined by:

δτ

σGk = − δp

σp→ σG = −δτ

δpσp = A

σpp0

(7.12)

For the SQM case, we assume no diffraction in time: the wave function willbe clipped by the gate, not diffracted. This means we must multiply the wavefunction by the gate function:

ϕG(+) (δτ) = e−(δτ)2

2W2 ϕG(−) (δτ) (7.13)

Now we convert back to p space, but on the far side of the gate:

ˆϕG(+) (p) = 4

√1

πσ2p

e− δp

2

2σ2p− A2

2W2δp2

p20−ı p

2

2m τG (7.14)

Evolve to detector:

ˆϕD (p) = 4

√1

πσ2p

e− δp

2

2σ2p− A2

2Σ2δp2

p20−ı p

2

2m τD (7.15)

Switch back to τ space in the same way as at the gate. The significant changeis from σp → σ′p where the primed dispersion in p is one-half the harmonic meanof the original dispersion in p and the width of the gate:

1

(σ′p)2 =

1

σ2p

+A2

W 2

1

p20

(7.16)

so we can write out the wave function at the detector by inspection, first inp space:

45

ˆϕD (p) = 4

√1

πσ2p

e− δp2

(σ′p)2−ıp2

2m τD (7.17)

then in τ space:

ϕD (δτ) = 4

√1

πσ2G

e− δτ2

(σ′G)2−ımL2

2T

(1− δτT + δτ2

T2

)(7.18)

With the primed dispersion in clock time given by:

σ′G ≡ Tσ′pp0

(7.19)

The dispersion in clock time at the detector is basically a scaled version ofthe dispersion in momentum post gate ( on the previous page).

Note the factor 4

√1

πσ2G

is unchanged. The ratio:(σ′GσG

)2

(7.20)

tells us what percentage of the particles get through the gate. But this doesnot effect the calculation of the dispersion, which is normalized. With the gatewidth in momentum space defined as:

Wp ≡Wp0

A(7.21)

We have have the effective dispersion of the wave function in clock time as:

(σ′G)2

=T 2

p20

(σ2pW

2p

σ2p +W 2

p

)(7.22)

When the gate is wide open, the dispersion at the detector is defined by theinitial dispersion of the beam:

Wp →∞⇒ σ′G → σp (7.23)

But when the gate is much narrow than the beam, the dispersion at thedetector is defined by the width of the gate:

Wp → 0⇒ σ′G →Wp (7.24)

The key point is that the narrower the gate, the narrower the beam. Whilethe approximations we have used have been simple, this is a fundamental im-plication of the SQM view of time, of time as classical. In SQM, wave functionsare not, by assumption, diffracted by a gate, they are clipped. And thereforetheir dispersion in time must be reduced by the gate rather than increased byit.

46

Source Gate Detector

W

Clipped

Di�racted

Di�racted

Clipped

Figure 7.2: Single slit in time with quantization in time

7.2 Single slit in time with quantization in timeFor TQ, we start with a wave function factored in time and momentum:

ψτ (t) = ϕτ (t) ˆϕτ (p) (7.25)

We are treating the p part as a carrier, using the wave function from above.The momentum part will be unchanged throughout.

By assumption, the gate will act only on the time part:

Gt = e−(t−A)2

2W2 (7.26)

We take the particle as starting with t0 = 0, and E0 such that E0

m � 1. Thewave function pre-gate is then:

ϕτ (t) = 4

√1

πσ2t

√1

f(t)τ

e−ıE0t+ı

E20

2m τ−1

2σ2t f

(t)τ

(t−t0−E0m τ)

2

(7.27)

On arrival at the gate it is:

ϕG(−) (tG) = 4

√1

πσ2t

√1

f(t)G

exp

−ıE0tG −(tG −A)

2

2σ2t

(1− ı A

mσ2t

) + ıp2

0

2mA

(7.28)

Post gate:

ϕG(+) (tG) = e−(tG−A)2

2W 2 ϕG(−) (tG) (7.29)

47

The effect of the gate on the wave function is to rescale it:

1

2σ2t

(1− ı A

mσ2t

) → 1

2W 2+

1

2σ2t

(1− ı A

mσ2t

) (7.30)

We define rescaling functions g2 and h by:

1

W 2+

1

σ2t

(1− ı A

mσ2t

) =1

g2σ2t

(1− ı h

mσ2t

) (7.31)

We have by inspection:

W → 0⇒ g → W

σt, h→ 0

W →∞⇒ g → 1, h→ A

(7.32)

In between these limits, we equate real and imaginary parts to get:

g2 (W,σ,A) =W 2

σ2t

(A2

m2 + σ2t (σ2

t +W 2))

(A2

m2 + (σ2t +W 2)

2) (7.33)

h (W,σ,A) = Aσ2tW

2

A2

m2 + σ4t + σ2

tW2

(7.34)

This in turn gives us the t wave function at the detector. We replace σt andτG in the initial wave function by g and h:

ϕD (tD) = N 4

√1

πσ2t

√1

fGe−ıE0tD− 1

2g2σ2t f′D

(tD−tD)2+ıp20

2m τD (7.35)

f ′D ≡ 1− ıh+ (T −A)

mσ2t

As the gate opens up entirely, as W → ∞, we recover the wave functionwith no gate, as expected:

g → 1h+ T −A→ Tϕ′D → ϕD

(7.36)

But as W → 0 we have:

σtg →Wh→ 0

(7.37)

Effectively the post gate wave function looks like an initial wave function withwidth in time A; the gate resets the wave function. We can reapply the analysisfrom above ( on page 34) to get:

48

(∆t)2

=

(1

2W 2+

1

2σ2τ

)τ2

m2(7.38)

Therefore:

1. In SQM the uncertainty in clock time is proportional to the width of thegate.

2. In TQ the uncertainty in clock time is inversely proportional to the widthof the gate.

These effects are results of the fundamental assumptions. In SQM, time is aparameter and the gate must clip the incoming wave function. In TQ, time isan operator and the gate must diffract the incoming wave function.

Therefore the observable effect may be made, in principle, arbitrarily large.

7.3 Double slit in time

Sources Detector

T

= T

Figure 7.3: Double slit

The double slit experiment has been described by Feynman [26] as the “cen-tral mystery” of quantum mechanics. It is less useful here: both TQ and SQMgive the same interference patterns at the detector.

We start with the analysis in Gerhard and Paulus [30]. They consider asetup where two Gaussian wave packets are “born” at the same position, butone at a time ∆T after the other. They consider wave packets with zero groupvelocity; we consider packets with group velocity p0

m .

49

We assume the source emits a Gaussian wave packet at time zero then attime ∆T later emits a second Gaussian wave packet with the same shape butwith overall phase Φ relative to the first.

Focusing on the time dependence, the wave packets may be written in mo-mentum space as:

ˆϕ0 (p) e−ıp2

2m τ (7.39)

and:

ˆϕ∆T (p) e−ıp2

2m (τ−∆T )−ıΦ (7.40)

The initial relative phase between the two is:

eıp20

2m∆T−ıΦ (7.41)

But – per the discussion earlier on page 31 the dependence of the phase on clocktime for each is the same:

e−ıp2

2m τ (7.42)

Therefore there is no additional interference beyond that resulting from theinitial phase difference, already accounted for in the analysis using SQM.

The interference pattern will therefore have the same spacing. Each of thepeaks will be a bit wider, but that effect is already accounted for in the analysisof the single slit.

It is curious that it is the single and not the double slit which provides thedecisive test between TQ and SQM.

8 Multiple particles

8.1 Why look at the multiple particle case?“In what follows we assume that even though A = δm is formally divergent, itis still “small” in the sense that it is of the order of 1/137 times the electronmass.” p271 of [72]

In principle, we have enough to test TQ. Why then look at the multipleparticle case?

From the above we see the effects of TQ will be largest at relativistic veloc-ities and short times. This implies we need to look at high energies and, sincehigh energies imply particle creation, at the multiple particle case5.

And there is the further concern that TQ may not be renormalizable. If theloop integrals in SQM are barely renormalizable, with the addition of one moredimension the loop integrals in TQ may well become completely intractable.

5We thank Dr. Steve Libby for bringing this point to our attention, personal communica-tion 2010

50

And extending TQ to the multiple particle case opens up some new effects,e.g. wave functions which are anti-symmetric in time (below on page 56).

And then there is of course the intrinsic interest of the question.At the same time, there are formidable difficulties: the literature on quantum

field theory is vast and complex (among the references we have found helpfulare [12, 13, 71, 70, 68, 43, 67, 86, 40, 87, 41, 54, 89, 21, 51, 39]). There is nopossibility of covering all possible cases. But we will work up enough kit thatwe can see TQ can be extended to include field theory with no worse that thedifficulties that are inevitable given that we have one more dimension in play.

To begin, consider the Laplace transform of the single particle propagator:

Ks (p) =2ım

p2 −m2 + 2ıms(8.1)

Compare this to the Feynman propagator for a spinless particle in SQM:

ı

p2 −m2 + iε(8.2)

Since taking the limit s→ 0 in Laplace transforms corresponds (loosely) toexploring longer times, τ → ∞, and longer times to the realm in which TQ isexpected to match SQM, we have a natural correspondence between the twoas6:

lims→0Ks (p) ∼ ˆKε (p) (8.3)

This suggests we can get a first take on the problem by:

1. Extruding the usual Fock space from three to four dimensions, so that allwave functions are extruded from three to four dimensions.

2. In any set of Feynman diagrams, replace the SQM propagators with TQpropagators.

3. Leave the topology of the diagrams unchanged.

4. Focus on experimental setups that show time dependence at start andfinish: so go from looking at beams and rates to looking at time of arrivaland the like.

This is very nearly what we will actually do here. To give a bit more depth,we will pin down the TQ Lagrangian by requiring that the resulting propagatormatch the single particle propagator above. This will give us the same topologyfor the Feynman diagrams, but also give us a Lagrangian that let us in principleexplore problems that are outside the scope of Feynman diagrams.

Since the complications of spin are inessential to the problem here, we willlook at a simple model with two massive spinless particles A and B (correspond-ingly very loosely to the electron and photon). This will let us work out the

6The factor of 2m in the numerator will drop out when we do the calculation more carefullybelow, essentially for dimensional reasons.

51

Lagrangian, the rules for Feynman diagrams, verify that TQ matches SQM inthe appropriate limit, and verify that in fact the loop diagrams are renormaliz-able ( on page 68).

In fact they are not merely renormalizable, they are convergent by inspection.As noted in the introduction, and as we will show, it appears that the familiarultraviolet divergences may be seen as side effects of assuming that time is flat. Ifthe location of a particle in time is certain, it has by the Heisenberg uncertaintyprinciple an infinite uncertainty in energy, and this infinity folds itself into theloop integrals, making them divergent.

8.2 ABC model

A

A’ B

Figure 8.1: Vertex interactions in the AB model

We will look at a simple quantum field model with two spinless, massivespecies of particles A and B.

A and B are models for an electron interacting with a photon, so we takethe mass m of A as of order me and the mass µ of B as of the order of the upperbound on the mass of the photon or 10−18eV [7]. A’s can emit and absorb B’s.

These are complex fields.We use straight lines for A, wavy for B, in view of their roles as proxies for

the electron and photon.

8.3 Fock spaceIn the single particle case we generalized the paths from three dimensions tofour. Here we generalize the wave functions that support Fock space likewise.

We use box normalization. The box runs from −L → L in all four coordi-nates, especially coordinate time. It is taken to extend well past the waveletswe are working with. If the particles are traveling from clock time zero to clocktime T, we will need to take L� T so that all interesting paths will be includedinside the box.

Using box normalization means that the Fourier transforms will be discrete.Discrete Fourier transforms are convenient for discussing various points of prin-ciple and to help in visualizing the field theory calculations. For the actualcalculations we will use continuous wave functions.

52

8.3.1 Fock space in SQM

Single particle basis wave functions We will focus on the x coordinatehere; y and z are the same.

We break our box into 2M pieces, implying a lattice spacing a as a ≡ LM . a

has dimensions of length.This lets us replace the smoothly varying x with the values of x at a series

of points:

x ∈ [−L,L]→ x = (−aM, . . . , 0 . . . , aM) (8.4)

With this approach, we have x = ai. We are using i for the index, ı for thesquare root of -1.

The general translation table is:

(i, j, k)↔ 1

a(x, y, z) (8.5)

Integrals over space go to sums over i, j, k:

L∫−L

dxdydz ↔ a3M∑

i,j,k=−M

(8.6)

We can simplify by representing the sum over i, j, k by a sum over −→x :

M∑i,j,k=−M

→∑~x

(8.7)

The coordinate basis is trivial, just Kronecker delta functions.

φ~x′ (~x) ≡ δ3 (~x− ~x′)→ δii′δjj′δkk′ (8.8)

We assume the wave functions are periodic in 2L; this gives allowed valuesof k.

φ (−L) = φ (L)→ −kiL = kiL+ 2πı→ kx =L

πi (8.9)

The periodic condition is not important; L will be chosen large enough thatall interesting wave functions are well inside of it.

We can simplify sums over these kx, ky, kz by writing as:

M∑kx,ky,kz=−M

→∑~k

(8.10)

We normalize the basis wave functions to one:

M∑i,j,k=−M

φ∗~k (~x)φ~k′ (~x) = δ~k~k′ (8.11)

53

Giving:

φ~k (~x) =1√

2L3 exp

(ı~k · ~x

)(8.12)

Now we can expand an arbitrary wave function in terms of the basis func-tions:

φ (~x) =∑~k

c~kφ~k (~x) (8.13)

The measure in the path integrals is is in terms of the c’s, not the basisfunctions. Fields complex, so have integrals over real and imaginary parts.

Dnφ =∏~k

dc~kdc∗~k

(8.14)

That is the integral at one step in time: we add an index n for the clocktime:

Dφ =

N∏n=0

Dnφ (8.15)

At the end we let M go to infinity. And then let L go to infinity.

Normalization in continuum limit To match the normalization of the La-grangian, the continuous wave functions need to have dimensions mass. We canget this by requiring they are normalized to 1√

2m:∫

d~xφ∗~k′ (~x)φ~k (~x) =1

2mδ(~k′ − ~k

)(8.16)

Therefore in the continuum limit we have:

φ~k (~x)→ 1√2m

1√

2π3 exp

(ı~k · ~x

)(8.17)

Multi-particle wave functions Fock space becomes interesting when we goto multiple particle wave functions. For two particles:

φ~k~k′ (1, 2) ≡ 1√2

(φ~k (1)φ~k′ (2) + φ~k (2)φ~k′ (1)

)(8.18)

We can use the familiar creation and annihilation operators to build thesewave function up in the usual way:

a†~k

∣∣n~k⟩ =√n~k + 1

∣∣n~k + 1⟩

a~k∣∣n~k⟩ =

√n~k∣∣n~k − 1

⟩ (8.19)

54

This implies the creation and annihilation operators have the usual commu-tation relations: [

a~k, a†~k′

]= δ~k~k′ (8.20)

Note we are not defining the creation and annihilation operators in termsof an infinite set of harmonic oscillators; we are defining them by their effectson Fock space. We can think of movement in Fock space as a giant game ofsnakes and ladders, with the creation operators as the ladders, the annihilationoperators as the snakes.

8.3.2 Fock space in TQ

Single particle basis wave functions The development in TQ works alongthe same lines but with one more dimension t and a fourth index h:

i, j, k → h, i, j, k (8.21)

t = ah (8.22)

Note there is no requirement that the lattice spacing a in coordinate time tmatch the step spacing ε in clock time τ ; in fact they will in general not:

a 6= ε (8.23)

We promote integrals and sums over three dimensions to integrals and sumsover four: ∫

d~x→∫dtd~x,

∑ijk

fijk →∑hijk

fhijk (8.24)

The coordinate basis is again given by Kronecker δ functions:

φx′ (x) ≡ δ4 (x− x′)→ δhh′δii′δjj′δkk′ (8.25)

And the basis functions in k space are now four dimensional:

φk (x) =1

4L2exp (−ıkx) (8.26)

The k’s are again periodic. Because we have been careful to make the boxwe are using much much greater than the times or distances of interest in ourspecific problem, we do not need to worry about the periodic condition in time.If we start our wavelets at clock time zero, and take their dispersions σt � Lthey will not overlap the ends of the box in time any more than they do inspace. 7

Again as a short cut label sums as sums over x or k:7It is not that the system in question did not exist before clock time \tau=0, it is that we

are assuming we can leave this point out of the analysis.

55

M∑h,i,j,k=−M

→∑x

M∑w,kx,ky,kz=−M

→∑k

(8.27)

Normalization in continuum limit With box normalization, the fields havedimension of m2. To match with the conventional normalization of the La-grangian, they need to have dimensions m

32 . Can get this by again requiring

they are normalized to 1√2m

:∫d4xφ∗k′ (x)φk (x) =

1

2mδ (w′ − w′) δ

(~k′ − ~k

)(8.28)

Therefore again add an additional factor of 1√2m

:

φk (x)→ 1√2m

1√

2π4 exp

(−ıwt+ ι~k · ~x

)(8.29)

Multiple particle wave functions Symmetrization in four dimensions worksexactly as in three. For two particles we have:

φkk′ (1, 2) ≡ 1√2

(φk (1)φk′ (2) + φk (2)φk′ (1)) (8.30)

And the creation and annihilation operators work in the same way:

a†k |nk〉 =√nk + 1 |nk + 1〉

ak |nk〉 =√nk |nk − 1〉

(8.31)

With the commutators: [ak, a

†k′

]= δkk′ (8.32)

This defines the Fock space in four dimensions, along the same lines as theone in three. We are again playing snakes and ladders, but in four dimensionsrather than three.

8.3.3 Anti-symmetry in time

We assume the same overall symmetry properties are required in four dimensionsas in three. This implies that wave functions can use the coordinate time to helpmeet their symmetry responsibilities, with potentially amusing implications. Inparticular if the wave function is anti-symmetric in time, it will have the “wrong”symmetry properties in space.

This is testable, at least in principle.

56

Say we have wide wave functions in time and space A (t) and B (x), andnarrow wave functions in time and space a (t) and b (x). The particles areidentified as 1 and 2.

An acceptable initial wave function is:

ϕsym (1, 2) =1√2

(A1B1a2b2 +A2B2a1b1) (8.33)

This clearly has the right symmetry between particles 1 and 2.We wish to break this down into sums over products of wave functions in

time and space.The symmetrical basis functions in time and space are:

ϕsym (1, 2) =1√2

(A1a2 +A2a1)

ϕsym (1, 2) =1√2

(B1b2 +B2B1)

(8.34)

If we use these as a product we get:

ϕsym (1, 2) ϕsym (1, 2) =1

2(A1B1a2b2 +A1B2a2b1 +A2B1a1b2 +A2B2a1b1)

(8.35)where the middle two terms do not belong.The anti-symmetric basis functions in time and space are:

ϕanti (1, 2) =1√2

(A1a2 −A2a1)

ϕanti (1, 2) =1√2

(B1b2 −B2B1)

(8.36)

their product is:

ϕanti (1, 2) ϕanti (1, 2) =1

2(A1B1a2b2 −A1B2a2b1 −A2B1a1b2 +A2B2a1b1)

(8.37)and the sum of the completely symmetric and the completely anti-symmetric

gives the target wave function:

ϕsym (1, 2) = ϕsym (1, 2) ϕsym (1, 2) + ϕanti (1, 2) ϕanti (1, 2) (8.38)

To get a wave function which is completely symmetric in time and spacetogether from products of wave functions in time and in space separately weneed to include wave functions anti-symmetric in time and anti-symmetric inspace.

Obviously an experimental setup which picks off the completely anti-symmetricterm would be tricky both to design and to execute. If it gave a null result,

57

that could easily be attributed to problems with the experiment itself. So thisis unlikely to be useful as a way to falsify TQ.

But the possibility is certainly an amusing one and suggests that TQ –whether correct or not – may be a useful source of experimental ideas.

8.4 LagrangianTo our two initial requirements – covariance and matching of SQM in the longtime limit – we add a third, that the propagator in TQ in the multiple particlecase match the propagator in the single particle case. This will unambiguouslydefine the Lagrangian.

In SQM the spin 0 free Lagrangian is:

L [φ, φ∗] =1

2

∂φ∗

∂τ

∂φ

∂τ− 1

2∇φ∗∇φ− m2

2φ∗φ (8.39)

The action is the integral of this:

T∫0

dτd~xL[φ, φ

](8.40)

Typically we let the limits in (clock time) go to ±∞ (0 → −∞, T → ∞),usually somewhere near the end of the analysis. Here that would average outeffects of any dispersion in time. We will have to keep the total clock time finite.

x and t have to rotate into each other under a Lorentz transformation; ourmain rule. So we have:

L0 =1

2∂µφ

∗∂µφ− m2

2φ∗φ =

1

2∂tφ∗∂tφ−

1

2∇φ∗∇φ− m2

2φ∗φ (8.41)

To include the dependence on clock time we will need to include integralover clock time from 0 to T :

S0 =

T∫0

dτ

∫dtd~xL0 [ϕ, ϕ] (8.42)

We have the further requirement to match the single particle propagator.The obvious problem is that there is no longer any τ dependence in the La-grangian. With no dependence on clock time we can’t possibly reproduce theSchrödinger equation. What adjustment to the Lagrangian is needed to repro-duce the Schrödinger equation?

The Schrödinger equation has the form in momentum space:

2mıd

dτφ = −

(ω2 − ~k2 −m2

)φ (8.43)

We break the field out into real and imaginary parts:

58

φ = φR + ıφI (8.44)

In momentum space, the free part of the Lagrangian is now:

L0 [φR, φI ] = −ω2 − ~k2 −m2

2

(φ2R + φ2

I

)(8.45)

Define β by:

β ≡ w2 − ~k2 −m2 (8.46)

By matching real and imaginary parts we have:

2md

dτφR = −βφI

2md

dτφI = βφR

(8.47)

Or as a pair:

2md

dτ

(φRφI

)=

(0 −ββ 0

)(φRφI

)(8.48)

The solutions have the form:

φ(R)τ + iφ(I)

τ = exp

(−ı β

2mτ

)(φ

(R)0 + iφ

(I)0

)(8.49)

Achieved by the following system:

L0 = −1

2βφ2

R −1

2βφ2

I (8.50)

Lclock = m(φRφI − φRφI

)(8.51)

L = Lclock + L0 (8.52)

This gives the Euler-Lagrange equations:

d

dτ

δLδφR

=δLδφR

→ φI =β

2mφR

d

dτ

δLδφI

=δLδφI→ φR = − β

2mφI

(8.53)

We recast the Lagrangian in terms of φ∗, φ:

Lclock = mı(φ∗φ− φ∗φ

)(8.54)

or:

59

L [φ, φ∗] = mı(φ∗φ− φ∗φ

)− β

2φ∗φ

which gives the equivalent equations:

2mı∂

∂τφ = −βφ

2mı∂

∂τφ∗ = βφ∗

We have the full TQ Lagrangian:

L [φ, φ∗] = mı(φ∗φ− φ∗φ

)+

1

2∂µφ

∗∂µφ− m2

2φ∗φ− V (φ∗, φ) (8.55)

8.5 KernelsWith the Lagrangian defined, we can write the full kernel – at least formally —as:

KT ≡∫Dφ exp

ı T∫0

dτ

∫d4xL

[φ, φ∗, φ, φ∗

] (8.56)

The measure is defined as the values of the fields at successive times n:

Dφ ≡M∏n=1

∏x

dφ∗n (x) dφn (x) (8.57)

This notation conceals much complexity. To extract physics from this weneed to apply it to a wave function, an element of our Fock space. And thento get the amplitude to arrive at a particular stage compute the transitionamplitude to another element of our Fock space:

〈ψ′|∫Dφ exp

ı T∫0

dτ

∫d4xL

[φ, φ∗, φ, φ∗

] |ψ〉 (8.58)

We will use the generating function approach, as in [44, 89]. We add sourcefunctions J to the Lagrangian:

L[φ, φ∗, φ, φ∗

]→ L

[φ, φ∗, φ, φ∗

]+ Jτ [x]φ (x) + J∗τ [x]φ∗ (x) (8.59)

and then define the generating function Z[J ] as the value of the kernel sand-wiched between the ground state |0〉:

60

Z [J ] ≡ 〈0|∫Dφ exp

ı T∫0

dτL[φ, φ∗, φ, φ∗

]+ Jτ [x]φ (x) + J∗τ [x]φ∗ (x)

|0〉(8.60)

We can get various Green’s functions from this by taking derivatives withrespect to the source functions J and then evaluating at the point where thesource is zero. To divide out the disconnected diagrams we take the log of thegenerating function W [J ]:

Z [J ] = Z [0] exp (ıW [J ]) (8.61)

and take the derivatives of W with respect to J . For instance to get thetwo-point kernel from x0 → xτ we take:

K(2)τ ′τ (x′;x) =

δ2W [J ]

δJτ ′ (x′) δJτ (x)

∣∣∣∣Jτ′ ,Jτ=0

(8.62)

We can get any arbitrary Green’s function by taking such derivatives. Wehave therefore provided TQ with the machinery needed to construct – in prin-ciple – an arbitrary Feynman diagram.

8.6 PropagatorTo close the loop we need to derive the TQ propagator from the generatingfunction. We are following the derivation in Zee [89], expanding from four tofive variables. With this approach we can write W out as:

W [J ] = −1

2

∫dτdτ ′

∫d4xd4x′Jτ (x)D (x− x′) Jτ ′ (x′) (8.63)

where D is the solution of the equation:

(2mι

∂

∂τ−(∂µ∂µ +m2

))Dττ ′ (x− x′) = δ (τ − τ ′) δ(4) (x− x′)

Once we have solved for D we may identify it as the propagator.We have already solved this equation (above 4.8); its kernel is defined by:(

ı∂

∂τ−(∂µ∂

µ +m2)

2m

)Kττ ′ (x;x′) = δ (τ − τ ′) δ(4) (x− x′)

in momentum space:(ı∂

∂τ+

(p2 −m2

)2m

)Kττ ′ (p; p

′) = δ (τ − τ ′) δ(4) (p− p′)

61

and explicitly:

Kττ ′ (p; p′) = exp

(ιp2 −m2

2m(τ − τ ′)

)δ(4) (p− p′) θ (τ − τ ′)

From this we infer that the kernel for TQ in field theory is:

Dτ (p, p′) =1

2mexp

(ıp2 −m2

2mτ

)δ4 (p− p′) θ (τ)

The 12m is expected from the normalization of the wave functions. Recall we

can write a kernel in the single particle case case:

K ∼∑n

φnφ∗n exp (−ιEnτ)

and that we went from normalizing to one to normalizing to 12m when we

built Fock space:

φ′ =φ√2m

so we have:

D ∼∑n

φ′n (φ′)∗n exp (−ιEnτ) ∼ 1

2mK

We are assuming retarded boundary conditions θ (τ − τ ′). Given our in-terpretation of clock time as the observer’s time this is probably a bit morenatural.

Note however there is no logical problem with using advanced boundaryconditions. For instance, if we are analyzing a crime scene and trying to seewhere the bullet came from we naturally solve in terms of advanced boundaryconditions – the location, depth, and angle of the bullet hole – and infer fromthem the starting position and velocity of the bullet. Even though the boundaryconditions are advanced, there is no question but that the causal direction wentfrom sinister intent to trigger pull to tragic impact. The difference betweenretarded and advanced boundary conditions is merely a question as to how bestto analyze the problem at hand.

We can compute the advanced propagator immediately. Assuming τ > 0wave function in the future is given by:

ψτ = K(retarded)τ ψ0

The advanced kernel is defined as the one that goes from future to past:

ψ0 = K(advanced)τ ψτ

Combining these two ideas:

62

ψ0 = K(advanced)τ K(retarded)

τ ψ0

and by inspection in momentum space:

K(advanced)τ (p; p′) =

1

2mexp

(−ιp

2 −m2

2mτ

)δ(4) (p− p′) θ (−τ)

Note that positive frequencies in QED correspond to wave components withvalue of E > 0 in the p2 −m2 term; negative frequencies to wave componentswith E < 0. They are not directly related to the sign of the clock frequency f :

f ≡ −E2 − ~p2 −m2

2m

For now we use retarded boundary conditions.

8.7 Feynman diagramsContinuing with the generating function approach we consider the generatingfunction for SQM. Starting with a Lagrangian:

L[φ]

= L0

[φ]− V

(φ)

(8.64)

We can get the generating function by taking derivatives with respect tosource terms J :

Z[J]

= exp

(−∫dτd~xV

(δ

δJτ (~x)

))∫Dφ exp

(ι

∫dτd~xL0

[φ]− V

(φ)

+ J φ

)(8.65)

For example for the simple potential:

V (φ) = ιλφ4

4!(8.66)

we can get the generating function by using:

V

(δ

δJτ (~x)

)→ ιλ

4!

δ4

δJτ (~x)4 (8.67)

From the generating function we can derive the familiar Feynman rules bytaking the derivatives with respect to the sources, and then evaluating at J = 0.

To apply the same approach in TQ start with the TQ Lagrangian:

L [φ] = Lclock[φ, φ

]+ L0 [φ]− V (φ) (8.68)

then apply the coordinate space transformations t → τ and ~x → t, ~x or themomentum space transformations E → f and ~p → E, ~p. We can see this assplitting off the time variable in SQM, assigning it the role of “clock time”, then

63

replacing the three space dimensions with coordinate time plus the three spacedimensions.

We get:

Z [J ] = exp

(−∫dτdxV

(δ

δJτ (x)

))∫Dφ exp

(ι

∫dτdxL0 [φ]− V (φ) + Jφ

)(8.69)

In the generating function approach the L0 or Lclock+L0 generates the lineswhile the potential generates the vertexes. The advantage of the generatingfunction approach is that it makes clear that SQM and TQ have the sametopology. The zeroth, first, second and higher order diagrams will have thesame set of lines and vertexes, only the labeling will be different.

Therefore assume we have a set of Feynman diagrams in SQM and wish togenerate the corresponding Feynman diagrams in TQ. We:

1. start with the Feynman diagrams in SQM,

2. in the initial wave functions replace E → f and ~p → E, ~p (or t → τ and~x→ t, ~x if working in coordinate space),

3. if the initial distributions in t and E are not otherwise available estimatethem using the entropic method above,

4. replace all the SQM propagators with TQ propagators,

5. at each vertex, keep the force constant the same but replace δ (E − E′) δ(3) (~p− ~p′)with δ (f − f ′) δ(4) (p− p′),

6. at each vertex, integrate over all five variables, f and p,

7. and on the outgoing legs – at the point of detection – combine the clocktime τ and coordinate time t variables by convolution as above, the rulebeing that t = τ at detection but not prior.

We probably do not need more than the first or second order Feynman diagramsto falsify TQ, and those we could construct by hand. Still it is interesting thatit is possible to do this, even at this level of rigor.

8.8 MetricsMost measurements are of individual particles rather than of pairs. Thereforewe can focus on measurements associated with individual legs.

In TQ the topology is unchanged, but the wave functions on the legs arelikely to be distorted in various ways.

In momentum space we may write them as:

ψτ

(w,~k

)= ρτ

(w,~k

)exp

(ıθτ

(w,~k

))(8.70)

or in coordinate space as:

64

ψτ (t, ~x) = ρτ (t, ~x) exp (ıθτ (t, ~x)) (8.71)

We can classify experimental effects to look for by looking for differences inbetween these and the equivalent SQM wave functions:

ˆψτ

(~k)

= ˆρτ

(~k)

exp(ı ˆθτ

(~k))

ψτ (~x) = ρτ (~x) exp (ıθτ (~x))(8.72)

As noted the TQ wave functions will have greater dispersion in time andenergy. Time-of-arrival measurements will be able to pick out greater dispersionin time.

Because of the greater dispersion in time, TQ wave functions will see po-tentials a bit earlier and later than otherwise would be the case: forces of an-ticipation and regret. TQ wave functions can “see” potentials that SQM wavefunctions might miss. This means the total probability for scattering can changeunder some circumstances:∫

dEd~kρ(E,~k

)6=∫d~k ˆρ

(~k)

In principle, there can also be differences in average phase:

〈θ (t, ~x)〉 6=⟨θ (~x)

⟩(8.73)

although the analysis of the double slit experiment suggests these will beharder to find.

And one further variation is to look for per wave vector dispersion in time,especially in cases where the dispersion in time at each wave vector is not equalto the average:

σE

(~k)6= 〈σE〉 (8.74)

If we run a charged particle through a magnetic field, fractionating the wavefunction by velocity, we can look at the traces left on an array of detectors. Ingeneral the dispersion – and therefore the traces left on the detectors at eachposition in a detector array – should be narrower in SQM than in TQ. If thetrace looks like a cut left by a sword, the trace from SQM should be more likethat left by a rapier, from TQ more like that left by a saber.

8.9 Point correction to the massDirect comparisons of SQM and TQ can be a bit tricky because they targetdifferent domains. For instance in SQM we often focus on the S-matrix, lookingat transition elements like:

〈k′1, k′2, . . . | k1, k2, . . .〉 (8.75)

65

m2

m2

m2

m2

m2

m2

free one two three

Figure 8.2: Point correction to the mass

These are excellent for describing beams and cross-sections. But beams andcross-sections imply averages and averages kill. SQM is often focused on thelong time limit, but TQ is most interesting over shorter times.

To make a fair comparison we will take an alternate approach: we willcompare TQ to SQM on a simple problem, one where we can use the methodsappropriate to each. We will look at the effects of a m2

0 → m20 + δm2 correction.

8.9.1 Point correction to the mass in SQM

The corresponding term in the Lagrangian is:

V = −δm2

2φ∗φ (8.76)

We expect:

ˆK(0)

(p) =ι

E2 − ~p2 −m20

→ ˆK (p) =ι

E2 − ~p2 −m20 − δm2

(8.77)

We can write the series for the propagator by inspection:

ˆK = ˆK(0)− ι ˆK

(0)δm2 ˆK

(0)− ˆK

(0)δm2 ˆK

(0)δm2 ˆK

(0). . . (8.78)

This is a geometric series with sum:

ˆK = ˆK(0) 1

1 + i ˆK(0)δm2

(8.79)

Expanding the kernel we get the expected sum:

ˆK =ι

E2 − ~p2 −m20

1

1− δm2

E2−~p2−m20

=ι

E2 − ~p2 −m2(8.80)

8.9.2 Point correction to the mass in TQ

For TQ, the question is complicated by the clock term. If we change m0 →m0 + δm we get:

66

V = iδm(φ∗φ− φ∗φ

)− δm2

2φ∗φ (8.81)

The propagator will go to:

K(0)τ (p) =

1

2m0exp

(ιE2 − ~p2 −m2

0

2m0τ

)→ Kτ (p) =

1

2 (m0 + δm)exp

(ιE2 − ~p2 −m2

0 − δm2

2 (m0 + δm)τ

)(8.82)

In TQ it is more natural to work in terms of convolutions. If we confineourselves to the effects of the − δm

2

2 part we get:

Kτ (p) = K(0)τ (p)−ιδm2

τ∫0

dτ1K(0)τ−τ1 (p) K(0)

τ1 (p)−δm4

τ∫0

dτ1K(0)τ−τ1 (p)

τ1∫0

dτ2K(0)τ1−τ2 (p) K

(0)τ2 (p)+. . .

(8.83)We can solve in terms of the Laplace transform:

Ks (p) = K(0)s (p)− ιδm2

(K(0)s (p)

)2

− δm4(K(0)s (p)

)3

+ . . . (8.84)

This is again a geometric series:

Ks (p) = K(0)s (p)

1

1 + ιδm2K(0)s (p)

(8.85)

The Laplace transform of the propagator is given by:

K(0)s (p) ≡

∞∫0

dτ exp (−ιsτ) exp

(ιE2 − ~p2 −m2

0

2m0τ

)(8.86)

or (noting how the factors of 2m0 cancel):

K(0)s (p) =

ι

E2 − ~p2 −m20 − 2ım0s

(8.87)

We expand the Laplace transform of the kernel:

Ks (p) =2m0ι

E2 − ~p2 −m20 − 2ım0s

1

1− δm2

E2−~p2−m20−2ım0s

=ι

E2 − ~p2 −m20 − δm2 − 2ım0s

(8.88)and invert the Laplace transform to get:

Kτ (p) =1

2m0exp

(ιE2 − ~p2 −m2

2m0τ

)(8.89)

67

We see that E2 − ~p2 −m20 → E2 − ~p2 −m2, but that the overall 1

2m0scale

and the scale associated with the clock time taum0

have been left unchanged.The mass of a particle may be seen as an estimate of the extent to which

fluctuations off mass shell are allowed. As m → 0, fluctuations off the massshell will be more and more severely punished by the small denominator. For aphoton, they will be essentially forbidden: consistent with the intuitive notionof a photon as a particle that is frozen in time.

8.10 Loop correction to the mass

pμ

kμ

K21μ( ) k( )

pμ kμ

K21m( ) p k( )

K10m( ) p( )

pμ

K32m( ) p( )

ˆ0 p( )

0

1

2

3

Figure 8.3: Loop correction to the mass

The ultraviolet divergences in field theory have become notorious. In general,loop integrals are associated with divergences. In our simple AB model thereis an amplitude for an A to emit a B then reabsorb it. As is well known, thiswill look like a correction to the mass, taking m2

0 → m20 + δm2 as in the last

subsection. Unfortunately the integral for the δm2 correction is divergent. Ifthe A particle has four-momentum p and emits a B with four-momentum k, tocompute the amplitude associated with the loop we will need to integrate overall possible values of the intermediate k:

δm2 ∼∫d4k

i

(p− k)2 −m2

i

k2 − µ2(8.90)

which is logarithmically at large k:

δm2 ∼∫d4k

k4(8.91)

TQ has the same problem: as the Laplace transform of the kernel makes theclosest correspondence to the SQM kernel, the integrals might about the same:

68

δm2 ∼∫d4k

i

(p− k)2 −m2 − 2mιs

i

k2 − µ2 − 2µιs(8.92)

8.10.1 Loop integral in TQ

A more systematic approach is to start with an initial Gaussian test function,apply the appropriate kernels to it, integrating over the coordinates in turn. Forinstance, if we take ϕ0 (p) as a Gaussian test function in momentum, then thetwo sides of the loop correspond to momentum space kernels:

Iτ =

∫d4kd4pK(µ)

τ (k) K(m)τ (p− k) ϕ0 (p) (8.93)

with kernels:

K(m)τ (p− k) =

1

2mexp

ι (E − w)2 −

(~p− ~k

)2

−m2

2mτ

(8.94)

K(µ)τ (k) =

1

2µexp

(ιw2 − ~k2 − µ2

2µτ

)(8.95)

If we are computing the corrections due to the loop interaction, we will belooking at convolutions of the form:

τ3∫τ0

dτ2K(m)32 (p)

τ2∫τ0

dτ1

(K

(m)21 (p− k) K

(µ)21 (k)

)K

(m)10 (p) ϕ0 (p) (8.96)

which are most naturally solved as products of the Laplace transforms of therespective parts of the diagram. For the loop part this is:

Is ≡∞∫

0

dτ exp (−sτ)

∫d4k

(K(m)τ (p− k) K(µ)

τ (k))

(8.97)

This is quadratically divergent, much worse than in SQM:

Is =

∫d4k

i

2 (k2(m+ µ)− 2kµp+ µ (p2 −m(m+ µ− 2is)))(8.98)

We may reasonably hope to improve this by applying it to a Gaussian testfunction. The Gaussians will dominate any merely polynomial divergence. And,thanks to Morlet wavelet analysis, they are completely general. In practice theintegrals over the loop variables for a specific value of the clock time will bewell-defined. For instance, consider the time part of the loop integral:

69

Iτ =1

4µmItIxIy Iz exp

(−ιm+ µ

2τ

)(8.99)

with:

It ≡∫dwdE exp

(ιw2

2µτ

)exp

(ι(E − w)

2

2mτ

)√1

πσ2E

exp

(ιEt0 −

(E − E0)2

2σ2E

)(8.100)

and similarly for the other three coordinates.We recognize this as the usual coordinate kernel with the roles of mass and

clock time and of energy and time interchanged. As this is amusing and maypoint to some deeper symmetry, we present the details:

Consider:

ϕ2 (t′′) =

∫dt′dtK21 (t′′; t′) K10 (t′; t) ϕ0 (t) (8.101)

with the definitions:

K21 (t′′; t′) =

√ıM

2πτ21e−ıM

(t′−t)2

2τ

K10 (t′; t) =

√ıM

2πτ10e−ıM

(t′−t)2

2τ

ϕ0 (t) = 4

√1

πσ2t

e−ıE0(t−t0)− (t−t0)2

2σ2t

ϕτ (t′′) = 4

√1

πσ2t

√1

f(t)τ

e

−ıE0t′′

f(t)τ

− 1

2σ2t f

(t)τ

(t′′−t0)2+ı

E20

2Mf(t)τ

τ

f (t)τ = 1− ι τ

Mσ2t

(8.102)

We note that we get the familiar wave function from the loop integral withthe substitutions:

E0 → t0, E → t, σ2E → σ2

t , t0 → −E0, τ →M,m→ τ10, w → t′, µ→ τ21

t′′ = 0(8.103)

Except for three small prefactors: the two square roots and a factor ofexp (−ιE0t0).

Reversing the substitutions in ϕτ (t′′) we therefore have:

It =

√2πµ

ιτ

√2πm

ιτexp (ιt0E0) 4

√1

πσ2E

√1

f(t)m+µ

e− E2

0

2σ2Ef(t)m+µ

+ıt20

2τf(t)m+µ

(m+µ)

(8.104)

70

So the clock time goes into the sum of the two masses m + µ and so on.Aside from the curiosity value of this trick, it clearly demonstrates that theloop integral – for fixed clock time – is finite. As we would expect from thepresence of the Gaussian and its entanglement with the kernels.

But the Laplace transforms over the clock time, needed to build larger struc-tures, can still give trouble.

We can take advantage of the fact that the coordinate forms of the kernels inTQ are significantly simpler than the corresponding kernels in SQM. Considerthe loop integral we have been discussing, but in coordinate space. For this wedo not need the initial wave function, just the kernels:

Iτ (x′;x) ≡ K(m)τ (x′;x)K(µ)

τ (x′;x) (8.105)

with the kernels:

K(m)τ (x′;x) = − m2

4π2τ2exp

(− ım

2τ(x′′ − x′)2 − ım

2τ)

K(µ)τ (x′;x) = − µ2

4π2τ2exp

(− ıµ

2τ(x′′ − x′)2 − ıµ

2τ) (8.106)

The Laplace transform of the loop is:

Is (x′;x) =

∞∫0

dτ exp (−sτ)K(m)τ (x′;x)K(µ)

τ (x′;x) (8.107)

Or with extraneous detail abstracted out:

Is (x′;x) = c

∞∫0

dτ1

τ4exp (−sτ) exp

(−ιaτ − ι b

τ

)a =

m+ µ

2

b ≡ m+ µ

2(x′ − x)

2

c ≡ m2

4π2

µ2

4π2

(8.108)

with explicit Laplace transform:

Is (x′;x) = −c2(a− ιs)2

K3

(2√−ιb√ιa+ s

)(−ιb)3/2√

ιa+ s(8.109)

where K3 is the modified Bessel function of the second kind, order 3.Clearly a great deal of supported structure is required to make this meaning-

ful; at the same time we have been able to write this without either regularizationor renormalization.

71

8.10.2 Discussion

While we have not established that the results of corrections in TQ matchthose in SQM, we have established that loop integrals are well-behaved in TQ.Therefore TQ is not merely renormalizable it does not need to be renormalizedin the first place8.

It is the combination of using physically realistic wave functions at the startand entanglement in time which is largely responsible for this. The integral overE entangled the initial wave function with the kernels, pulling in the dispersionσ2E which ensured each integral in turn converged.Further as the effective cutoff is determined by the initial dispersion of the

wave function in time, it is subject – in principle – to experimental control.Therefore we could – in principle – look for dependence of loop corrections onthe dispersion in time/energy of the initial wave function.

With that we have completed the program laid out at the start of thissection we have shown how to do arbitrary calculations in field theory and wehave shown that the loop integrals are finite.

Obviously there is a great deal more work to be done to bring the work herein direct contact with experiment. Any realistic calculations are likely to bedifficult. But at least we have a starting point.

9 Discussion“It is difficult to see what one does not expect to see.” – William

Feller[24]

To return to our initial question, quite a few equations back, how do Alice andBob reconcile their respective wave functions?

To first order, given the long, slow approximation they will have:

ψ(Bob)

τBob

(x(Bob)

)≈ ψ

(Alice)

τAlice

(Λ

(Bob)(Alice)x

(Alice))

(9.1)

If they need more accuracy then this – if say they are investigating thelong slow approximation itself – they can determine the local rest frame of thevacuum and use that to define a shared clock time:

ψ(Bob)

τV acuum

(x(Bob)

)= ψ

(Alice)

τV acuum

(Λ

(Bob)(Alice)x

(Alice))

= ψ(V acuum)

τV acuum

(Λ

(Bob)(V acuum)x

(V acuum))

(9.2)Any experimental apparatus should be described in a fully four dimensional

way. The interactions of the apparatus with various quantum systems may bethought of as points where a classical description yields to a quantum one. Aliceand Bob may continue to think of themselves classically of course, even if theneurons they use for this must ultimately be modeled in quantum terms9.

8The mathematically correct expression is that in TQ the loop integrals do not need to beregularized. They still need to be normalized.

9We are not referring to Penrose’s hypothesis that neural interactions should be modeled

72

9.1 Is TQ falsifiable?To apply the rules of quantum mechanics along the time dimension:

1. We took path integrals as the defining representation. This made theextrusion from three to four dimensions unambiguous.

2. We used Morlet wavelet analysis rather than Fourier analysis to definethe initial wave functions. This let us avoid the use of unphysical wavefunctions and made achieving convergence and normalization of the pathintegrals possible.

3. We distinguished carefully between “clock time” and “coordinate time”.Clock time is a term that comes from the literature (often laboratory timeis used equivalently) and is defined operationally and therefore unambigu-ously. Coordinate time is the time dimension the paths move in. As itis defined by covariance, coordinate time’s properties are unambiguouslydefined by the properties of the space coordinates of the paths. If we usea certain measure or Lagrangian term with respect to space we must usethe same with respect to coordinate time.

As a result, once we have applied the requirement (via the long, slow approxima-tion) that TQ match SQM in the appropriate limit, there are no free parameters.

That does not fully establish that there is not another way to apply quantummechanics along the time dimension10. We have therefore been careful to focuson dimensional and symmetry arguments, which give first order predictionswhich are likely to be independent of the specifics of whatever method we mightuse. These first order predictions of TQ do not have much “give”:

• The initial dispersion in time is fixed by symmetry between time and spaceand the principle of maximum entropy.

• The evolution of the wave functions is fixed by the long, slow approxima-tion. This allows for give, but only over times of picoseconds, glacial bythe standards of TQ.

• We could choose Alice’s frame or Bob’s to do the analysis, but the correc-tions due to this are of second order. And in any case can be eliminatedby selecting “the rest frame of the vacuum” as the defining frame for TQ.

The predictions of TQ are therefore falsifiable to first order.

quantum mechanically, but more generally to the point that we would have no chemistry oreven stable atoms without quantum effects.

10Since the path integral and canonical quantization approaches are equivalent in SQM,one could for instance start with the canonical quantization approach and either posit someequivalent to the clock time/coordinate time split here or see if the canonical quantizationformalism does not suggest a new line of attack.

73

9.2 Experimental effectsWe have discussed four specific effects:

1. generally increased dispersion in time, as time-of-arrival effects.

2. the time/energy uncertainty principle, as the single slit in time.

3. anti-symmetry in time.

4. effects of loop cutoffs.

The first two are, obviously, the mostly important. Averaging can kill them, butthey should be present – and with sufficient cunning and patience observable –in any experimental setup in which the sources vary in time and the detectorsare time-sensitive.

Additionally we could look for:

• shadowing in time – self-interference by detectors (or sources).

• effects of time and energy being first class operators.

And more generally, any quantum effect seen in space is likely to be visiblefor time as well. TQ is to SQM with respect to time as SQM is to classicalmechanics with respect to space.

The experiments are likely to be difficult. The attosecond times we areprimarily interested in are at the edge of detectible. The investigation here waspartly inspired by Lindner’s “Attosecond Double-Slit Experiment” [53]. But thetimes there, 500as, are far too long for us. More recent work has reached shorterand shorter times: 12 attoseconds in Koke, Sebastian, and Grebing [49] and asnoted the extraordinary sub-attosecond times in Ossiander [65]. The effects ofdispersion in time should now be within experimental range.

Reviews of foundational experiments in quantum mechanics (for exampleLamoreaux [50], Ghose [31], and Auletta [10]11) provide a rich source of candi-date experiments: the single and double slit as well as many other foundationalexperiments have an “in time” variant, typically with time and a space dimensionflipped.

9.3 Extensions of TQWe have provided only a basic toolkit for TQ. Possible extensions include:

1. Generalizing the treatment of spinless bosons to include photons andfermions. Preliminary investigation suggests there is no problem of prin-ciple with extending the “clock term” approach to these cases.

2. Derivation of the bound state wave functions from first principles.11Auletta alone provides a list of about three hundred foundational experiments.

74

3. Scattering experiments. For instance, a probe particle with a narrow widthin time can act as the equivalent of a high speed single slit.

4. Exact treatment of the single slit in time, included paths that wander backand forth through the slit.

5. Careful treatment of measurements, including paths that overshoot, un-dershoot, and loop around the detector.

6. Decoherence in time.

7. Infrared divergences. From the point of view of TQ, these may be the flipside of the ultraviolet divergences, suppressed in a similar way.

8. More detailed treatment of the ultraviolet divergences. We have not forinstance ruled out the possibility that TQ is already ruled out by thespecifics of existing loop corrections.

9. The spin-statistics connection. Obviously the T in the CPT theorem wouldrequire re-examination.

10. Statistical mechanics. Implicit in TQ is the assumption that the initialsmooth wave function should be replaced by a statistical ensemble of fluc-tuations in time. And of course any statistical ensemble should includefluctuations in time.

11. Standard model.

12. More detailed treatment of the choice of frame.

13. And of course quantum gravity – since TQ is by construction highly sym-metric between time and space and as shown free of the ultraviolet diver-gences, it may be a useful starting point for attacks on the problem ofquantum gravity.

9.4 Alternatives to TQTQ is, by construction, something of a lowest common denominator for quantumeffects along the time dimension. The requirements of covariance, consistencywith the known, and falsifiability severely constrained its development. Wesuspect it may be more of a way station than a destination. Certainly its de-velopment was strongly informed by a number of nearby alternatives including:

• the relativistic dynamics program of Horwitz, Land, and others: [52, 32,1, 39, 2],

• the time symmetric quantum mechanics of Aharonov and Reznik: [6, 69],

• and Cramer’s transactional interpretation[18, 19, 45, 20].

75

9.5 If TQ is falsified?Let’s suppose that one or more of the proposed experiments is done and conclu-sively demonstrates that TQ is false. This would in turn raise some interestingquestions:

1. Is there a frame in which TQ is maximally (or minimally) falsified? Thatwould be a preferred frame, anathema to relativity (but see the rest frameof the vacuum above).

2. Is TQ equally false in all frames? then how do we reconcile the disparatewave functions of Alice and Bob?

As TQ was built by combining special relativity and quantum mechanics in arelatively straightforward way, experiments that falsify TQ may offer insight onone or both of those theories.

Appendices

A ConventionsWe use natural units ~ = c = 1. When summing over three dimensions weusually use i, j, k. We use ı sans dot for the square root of -1, i with dot for theindex variable.When summing over four dimensions we use h for the coordinatetime index so such sums run over h, i, j, k.

A.1 Time and space variables

We use the symbols w,~k and E, ~p for the momentum variables complemen-tary to coordinate time t and space ~x. We use τ for clock time and f for itscomplementary variable, clock frequency.

In particular:wop ≡ ı ∂∂t , E

op ≡ ı ∂∂tfop ≡ ı ∂∂τ

(A.1)

When there is a natural split into coordinate time and space parts we use atilde to mark the time part, an overbar to mark the space part. For example:

ψ (t, ~x) = ψτ (t) ψτ (~x) (A.2)

This is to reinforce the idea that in this analysis the three dimensional partis the average (hence overbar), while the coordinate time part contributes a bitof quantum fuzziness (tilde) on top of that.

We use an overdot to indicate the partial derivative with respect to labora-tory time:

fτ (t, ~x) ≡ ∂fτ (t, ~x)

∂τ(A.3)

76

A.2 Fourier transformsWe use a caret to indicate that a function or variable is being taken in momen-tum space. To keep the Fourier transform itself covariant we use opposite signsfor the coordinate time and space parts:

f (E, ~p) = 1√2π

4

∞∫−∞

dtd~xeıEt−ı~p·~xf (t, ~x)

f (t, ~x) = 1√2π

4

∞∫−∞

dEd~pe−ıEt+ı~p·~xf (E, ~p)(A.4)

For plane waves:

φp (x) = φ (t) φ (~x) = 1√2π

exp (−ıEt) 1√2π

3 exp (ı~p · ~x)

φx (p) =ˆφ (E) ˆφ (~p) = 1√

2πexp (ıEt) 1√

2π3 exp (−ı~p · ~x)

(A.5)

To shorten the expressions we use:

x ≡ (t, ~x) = (t, x, y, z)

The difference between x the four vector and x the first space coordinate isgenerally clear from context. In momentum space we use:

p ≡ (E, ~p) = (E, px, py, pz)

A.3 Laplace transformsWe define the Laplace transform in the usual way, for instance.

L [Kτ ] ≡∞∫

0

dτKτ exp (−sτ) (A.6)

We use script letters to indicate the Laplace transform of an object:

Ks ≡ L [Kτ ] (A.7)

B Classical equations of motionWe show that we get the classical equations of motion from the Lagrangian:

L (xµ, xµ) = −1

2m (xµxµ)− qxµAµ (x)− 1

2m (B.1)

Broken out into time and space parts the Lagrangian is:

L(t, ~x, t, ~x

)= −1

2mt2 +

1

2m~x · ~x− qtΦ (t, ~x) + qxjAj (t, ~x)− 1

2m (B.2)

77

The potentials are not themselves functions of the laboratory time τ . From theEuler-Lagrange equations we have:

mt = −qΦ + qtΦ,0 − qxjAj,0 = −qxj (Φ,j +Aj,0) (B.3)

mxi = −qAi − qtΦ,i + qxjAj,i = −qtAi,0 − qxjAi,j − qΦ,it+ qxjAj,i (B.4)

Here the Roman indexes, i and j, go from 1 to 3 and if present in pairs aresummed over. We use an overdot to indicate differentiation by the laboratorytime τ .

By using:

~E = −∇Φ− ∂ ~A

∂t(B.5)

~B = ∇× ~A (B.6)

we get:

mt = q ~E · ~x (B.7)

which describes work being done on/by the particle and:

m~x = qt ~E + q~x× ~B (B.8)

which are the equations of motion of a classical particle in an electromagneticfield.

C UnitaritySince there is a chance of the wave function “sneaking past” the plane of thepresent, we have to be particularly careful to confirm unitarity.

To establish that the path integral kernel is unitary we need to establishthat it preserves the normalization of the wave function. The analysis in thetext only established this for the free case. We therefore need to confirm thatthe normalization of the wave function is preserved in the general case. We usea proof from Merzbacher [57] but in four rather than three dimensions.

We form the probability:

P ≡∫d4xψ∗ (x)ψ (x) (C.1)

We therefore have for the rate of change of probability in time:

dP

dτ=

∫d4x

(ψ∗ (x)

dψ

dτ(x) +

dψ∗

dτ(x)ψ (x)

)(C.2)

The Schrödinger equations for the wave function and its complex conjugate are:

78

dψ

dτ=−ı2m

∂µ∂µψ +q

m(Aµ∂µ)ψ +

q

2m(∂µAµ)ψ + ı

q2

2mAµAµψ − ı

m

2ψ (C.3)

dψ∗

dτ=

ı

2m∂µ∂µψ

∗ +q

m(Aµ∂µ)ψ∗ +

q

2m(∂µAµ)ψ∗ − ı q

2

2mAµAµψ

∗ + ım

2ψ∗

(C.4)We rewrite dψ

dτ and dψ∗

dτ using these and throw out canceling terms. Since theprobability density is gauge independent, we choose the Lorentz gauge ∂A = 0to get:

dP

dτ=

∫d4x

(ψ∗(− ı

2m∂µ∂µψ +

q

m(A∂)ψ

)+( ı

2m∂µ∂µψ

∗ +q

m(A∂)ψ∗

)ψ)

(C.5)We integrate by parts; we are left with zero on the right:

dP

dτ= 0 (C.6)

Therefore the rate of change of probability is zero, as was to be shown. Andtherefore the normalization is correct in the general case.

D Gauge transformationsWe have all the usual possibilities for gauge transformations and in additionthe possibility of gauge transformations which are a function of the laboratorytime.

We can write the wave function as a product of a gauge function in coordinatetime, space, and laboratory time and a gauged wave function

ψ′τ (t, ~x) = eıqΛτ (t,~x)ψτ (t, ~x) (D.1)

If the original wave function satisfies a gauged Schrödinger equation:

(ıd

dτ− qAτ (x)

)ψτ (x) = − 1

2m

((p− qA)

2 −m2)ψτ (x) (D.2)

the gauged wave function also satisfies a gauged Schrödinger equation:

(ıd

dτ− qA′τ (x)

)ψ′τ (x) = − 1

2m

((p− qA′)2 −m2

)ψ′τ (x) (D.3)

provided we have:

79

A′τ (x) = Aτ (x)− dΛτ (x)

dτ(D.4)

and the usual gauge transformations:

A′µ = Aµ − ∂µΛτ (x) (D.5)

or:

Φ′ = Φ− ∂Λ∂t

~A′ = ~A+∇Λ(D.6)

If the gauge function Λ is not a function of the laboratory time (Λ = Λ (t, ~x))then we recover the usual gauge transformations for Φ and ~A.

In the text we do not take advantage of gauge functions that depend on theclock time, but it is a potentially amusing one: we could shift back and forthbetween gauges as the problem required.

E Free wave functions and kernelsIn general the free wave functions and kernels can be written as a coordinatetime part times a familiar non-relativistic part. The division into coordinatetime, space, and – occasionally – clock time parts is to some extent arbitrary.We make specific choices here but sometimes vary them in the text.

E.1 Plane wave functionsPlane wave in coordinate time:

φτ (t) =1√2π

exp

(−ıE0t+ ı

E20

2mτ

)(E.1)

Plane wave in space:

φτ (~x) =1√

2π3 exp

(ı~p0 · ~x− ı

~p02

2mτ

)(E.2)

The full plane wave is the product of coordinate time and space plane waves:

φτ (x) = φτ (t) φτ (~x) exp

(ım2

2mτ

)=

1

4π2exp (−ıp0x− ıf0τ) (E.3)

with definition of clock frequency:

f0 ≡ −E2

0 − ~p02 −m2

2m(E.4)

80

The equivalents in momentum space are δ functions with a clock time de-pendent phase:

ˆφτ (E) = δ (E − E0) exp

(ıE2

0

2mτ

)ˆφτ (~p) = δ(3) (~p− ~p0) exp

(−ı ~p

20

2mτ

)φτ (p) =

ˆφ0 (E) ˆφ0 (~p) exp

(−ım

2

2mτ

)= δ(4) (p− p0) exp (−ıf0τ)

(E.5)

E.2 Gaussian wave functionsE.2.1 Time and energy

Gaussian test function in coordinate time at clock time zero:

ϕ0 (t) ≡ 4

√1

πσ2t

e−ıE0(t−t0)− (t−t0)2

2σ2t (E.6)

ˆϕ0 (E) ≡ 4

√1

πσ2E

eıEt0− (E−E0)2

2σ2E (E.7)

With these conventions, the energy and coordinate time dispersions are re-ciprocals:

σE =1

σt(E.8)

Gaussian test function for coordinate time as a function of clock time:

ϕτ (t) = 4

√1

πσ2t

√1

f(t)τ

e−ıE0t+ı

E20

2m τ−1

2σ2t f

(t)τ

(t−t0−E0m τ)

2

(E.9)

with:

f (t)τ ≡ 1− ı τ

mσ2t

(E.10)

Alternate form:

ϕτ (t) = 4

√1

πσ2t

√1

f(t)τ

e

−ıE0t

f(t)τ

− 1

2σ2t f

(t)τ

(t−t0)2+ıE2

0

2mf(t)τ

τ

(E.11)

with expectation, probability density, and uncertainty (γ ≡ Em ):

81

〈t〉 = t0 + Emτ = t0 + v0τ = t0 + γτ

ρτ (t) =√

1πσ2

texp

− (t−γτ)2

σ2t

(1+ τ2

mσ2t

)

(∆t)2 ≡

⟨t2⟩− 〈t〉2 =

σ2t

2

∣∣∣1 + τ2

m2σ4t

∣∣∣(E.12)

In the non-relativistic case, if we start with t0 = τ , then we have 〈t〉 ≈ τthroughout.

For longer clock times the uncertainty in coordinate time is proportional tothe uncertainty in the energy:

∆t ∼ τ

mσt=

τ

mσE (E.13)

Gaussian test function in energy:

ˆϕτ (E) ≡ 4

√1

πσ2E

eıEt0− (E−E0)2

2σ2E

+ıE2

2m τ (E.14)

with expectation, probability density, and uncertainty:

〈E〉 = E0

ˆρτ (E) = ˆρ0 (E) =

√1

πσ2E

exp

(− (E − E0)

2

σ2E

)

(∆E)2

=σ2E

2

(E.15)

E.2.2 Single space/momentum dimension

Gaussian test function in one space dimension at clock time zero:

ϕ0 (x) = 4

√1

πσ2x

eıp0(x−x0)− (x−x0)2

2σ2x (E.16)

and in momentum:

ˆϕ0 (p) = 4

√1

πσ2p

e−ıpx0− (p−p0)2

2σ2p (E.17)

The space and momentum dispersions are reciprocal:

σp ≡1

σx(E.18)

Gaussian test function in one space dimension as a function of clock time:

82

ϕτ (x) = 4

√1

πσ2x

√1

f(x)τ

eıp0x− 1

2σ2xf

(x)τ

(x−x0− p0m τ)

2−ı p20

2m τ (E.19)

The definition of f (x)τ is parallel to that for coordinate time (but with the

opposite sign for the imaginary part):

f (x)τ = 1 + ı

τ

mσ2x

(E.20)

Alternate form:

ϕτ (x) = 4

√1

πσ2x

√1

f(x)τ

eıp0x

f(x)τ

− 1

2σ2xf

(x)τ

(x−x0)2−ı p20

2mf(x)τ

τ(E.21)

with expectation, probability density, and uncertainty for x:

〈x〉 = x0 + pxm τ = x0 + vxτ

ρτ (x) =√

1πσ2

1exp

(− (x−〈xτ 〉)2

σ2x

)(∆x)

2 ≡⟨x2τ

⟩− 〈xτ 〉2 =

σ2x

2

∣∣∣1 + τ2

m2σ4x

∣∣∣ (E.22)

and similarly for y and z.As clock time goes to infinity, the dispersion in space scales as:

(∆x)2 ≈ τ2

2m2σ2x

(E.23)

As noted, in the non-relativistic case the expectation of t is a proxy for theclock time 〈t〉 ∼ τ .

Negative x-momentum is movement to the left, positive to the right.As we require the most complete parallelism between time and space, we

therefore have that positive energy corresponds to movement into the future,negative into the past. As most of our wave functions have an energy of order:

E ∼ m+~p2

2m� 0 (E.24)

they are usually going into the future. As expected.Gaussian test function for momentum in one dimension as a function of clock

time:

ˆϕτ (p) = 4

√1

πσ2p

e−ıpx0− (p−p0)2

2σ2p−ı p

2

2m τ (E.25)

and expectation, probability density, and uncertainty for p:

83

〈p〉τ = 〈p〉0 = p0

ˆρτ (p) = ˆρ0 (p) =

√1

πσ2p

exp

(− (p− p0)

2

σ2p

)

(∆p)2

=σ2p

2

(E.26)

E.2.3 Four space

Usually we treat the case of four dimensions as a simple product of the cases ofthe individual dimensions. But it is more appropriate in general to treat themas a single covariant object.

We define the four dimensional dispersion Σ in position space at clock timezero:

Σµν0 ≡

σ2t 0 0 0

0 σ2x 0 0

0 0 σ2y 0

0 0 0 σ2z

(E.27)

With this we have the wave function at clock time zero:

ϕ0 (x) = 4

√1

π4det (Σ0)e−ıpµ0 (x−x0)µ−

12 Σµντ (x−x0)

µ(x−x0)

ν (E.28)

Four dimensional dispersion as a function of clock time τ :

Σµντ ≡

σ2t f

(t)τ = σ2

t − ı τm 0 0 0

0 σ2xf

(x)τ = σ2

x + ı τm 0 0

0 0 σ2yf

(y)τ = σ2

y + ı τm 0

0 0 0 σ2zf

(z)τ = σ2

z + ı τm

(E.29)

Four dimensional wave function as a function of clock time:

ϕτ (x) = 4

√det (Στ )

π4det2 (Στ )e−ıpµ0 (x−x0−vτ)µ−

12 Σµντ (x−x0−vτ)

µ(x−x0−vτ)

ν−ıf0τ

(E.30)with the obvious definition of the four velocity:

vµ ≡pµm

(E.31)

In the general case, the wave function does not, of course, split into coordi-nate time and space factors. But it may still be represented as a sum over basiswave functions that do, via Morlet wavelet decomposition.

84

And in momentum space at clock time zero:

ϕ0 (p) = 4

√√√√ 1

πdet2(

Σ0

)e−ıpx0− 12 Σµντ (p−p0)µ(p−p0)ν (E.32)

and as a function of clock time:

ϕτ (p) = 4

√√√√ 1

πdet2(

Σ0

)e−ıpx0− 12 Σµντ (p−p0)µ(p−p0)ν−ıω0τ = ϕ0 (p) exp (−ıf0τ)

(E.33)We define the momentum dispersion matrix Σ as the reciprocal of the coor-

dinate space dispersion matrix Σ:

Στ ≡ Σ−1τ (E.34)

E.3 Free kernelsWe list the kernels corresponding to the Schrödinger equation (4.8). Theseare retarded kernels going from clock time zero to clock time τ , so include animplicit θ (τ).

E.3.1 Coordinate time and space

Kernel in coordinate time:

Kτ (t′′; t′) =

√ım

2πτe−ım

(t′′−t′)2

2τ (E.35)

In three space we have the familiar non-relativistic kernel (e.g. [57]):

Kτ (~x′′; ~x′) =

√− ı

2πτ

3

exp

(ım

(~x′′ − ~x′)2

2τ

)(E.36)

In three space we have the familiar non-relativistic kernel (e.g. [57]):

Kτ (~x′′; ~x′) =

√− ı

2πτ

3

exp

(ım

(~x′′ − ~x′)2

2τ

)(E.37)

The full kernel is:

Kτ (x′′;x′) = Kτ (t′′; t′) Kτ (~x′′; ~x′) exp(−ım

2τ)

(E.38)

Explicitly:

Kτ (x′′;x′) = − m2

4π2τ2exp

(− ım

2τ(x′′ − x′)2 − ım

2τ)

(E.39)

85

With Laplace transform:

Ks (x) = −m2

√2s+ ım

2π2

√− ı

mx2K1

(√im+ 2s√− ımx2

)(E.40)

x→ x′′ − x′ (E.41)

where K1 is the modified Bessel function of the second kind.

E.3.2 Momentum space

Energy part:

ˆKτ (E′′;E′) = δ (E′′ − E′) exp

(ıE′

2 −m2

2mτ

)(E.42)

Three momentum part:

~Kτ (~p′′; ~p′) = δ(3) (~p′′ − ~p′) exp

(−ı (~p

′)2

2mτ

)(E.43)

Again, the same as non-relativistic kernel.The full kernel is:

Kτ (p′′ − p′) = ˆKτ (E′′;E′) ~Kτ (~p′′; ~p′) (E.44)

Spelled out:

Kτ (p′′; p′) = δ(4) (p′′ − p′) exp

(ıE′

2 − (~p′)2 −m2

2mτ

)(E.45)

or:

Kτ (p′′; p′) = δ(4) (p′′ − p′) exp (−ıfp′τ) (E.46)

With definition of clock frequency as above:

fp ≡ −E2 − ~p2 −m2

2m(E.47)

And Laplace transform:

Ks (p) =2ım

p2 −m2 + 2ıms(E.48)

86

F AcknowledgmentsI thank my long time friend Jonathan Smith for invaluable encouragement,guidance, and practical assistance.

I thank Ferne Cohen Welch for extraordinary moral and practical support.I thank Martin Land, Larry Horwitz, and the organizers of the IARD 2018

Conference for encouragement, useful discussions, and hosting a talk on thispaper at IARD 2018.

I thank Y. S. Kim for organizing the invaluable Feynman Festivals, for severalconversations, and for general encouragement and good advice.

I thank the experimenter at one of the Feynman Festivals who provided theprinciple motivation for the approach here (as noted in the introduction).

I thank the anonymous reviewers at the online journal Quanta, especiallyfor their comments on the need for an emphasis on testable and on clarity.

I thank Catherine Asaro, J. Barbour, Gary Bowson, Howard Brandt, H.Brown, John Cramer, J. Ferret, Robert Forward, N. Gisin, Fred Herz, J. Peřina,A. Khrennikov, David Kratz, Steve Libby, Andy Love, O. Maroney, John Myers,Paul Nahin, Marilyn Noz, R. Penrose, Stewart Personick, V. Petkov, H. Price,Matt Riesen, Terry Roberts, J. H. Samson, L. Smolin, L. Skála, R. Tumulka,J. Vaccaro, L. Vaidman, A. Vourdas, H. Yadsan-Appleby, S. Weinstein, and A.Zeilinger for helpful conversations

I thank the organizers of the Feynman Festival, QUIST, DARPA, and Perime-ter Institute conferences.

And of course, none of the above are in any way responsible for any errorsof commission or omission in this work.

References[1] International association for relativistic dynamics. 2014. URL http://

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