The Klein-Gordon Equation Revisited

Ken WhartonAssociate Professor

Department of Physics San José State University

San José, CA; USA

PIAF-1

February 1-3, 2008

Sydney, Australia

The PIAF Connection

• I will outline a foundational research program that naturally links to very different work by at least three PIAF participants:– Robert Spekkens– Huw Price– Lucien Hardy

• If successful, this program should also interest: – PI’s quantum gravity experts– Australia’s retrocausal experts– Bayesians (?)

The Big Picture(for neutral, spinless fields)

GeneralRelativity

SpecialRelativity

Non-Relativis

ticLimit

Classical Quantum

Quantum Gravity

Klein-GordonEquation

SchrödingerEquation

Quantum Mechanics

Quantum Field

Theory

Klein-Gordonin curved space

??????

A

B

C

Schrödinger’s starting point:The Klein-Gordon Equation (KGE)

Advantages: Time-Symmetric, Relativistically CovariantProblem: No consistent, spatially meaningful interpretation

€

φ(r, t) = a(k)e i(k⋅r−ωt ) + b(k)e i(k⋅r+ωt )dk∫

€

ω(k) = k 2c 2 + m2c 4 /h2

deBroglie Waves:

€

φ(x, t) ~ cos(k ⋅x −ωt) ; E = hω,p = hk

€

E 2 − p2c 2 − m2c 4 = 0

€

h2 ∂ 2

∂t 2− c 2h2∇ 2 + m2c 4

⎛

⎝ ⎜

⎞

⎠ ⎟φ(r, t) = 0

Relativistic Particle Klein-Gordon Equation (KGE)

General solutions to KGE:

What happens in the non-relativistic limit?

€

φ(r, t) ≅ e−iω0ta(k)e i(k⋅r−ω1t ) + e+iω0tb(k)e i(k⋅r+ω1t )dk∫

One does NOT get the Schrödinger equation!(1st and 2nd order differential equations

aren’t equivalent in ANY limit.)

€

ω(k) = k 2c 2 + m2c 4 /h2

€

ω ≅mc 2

h+

hk 2

2m≡ ω0 + ω1(k)

Schrödinger’s critical assumption:

€

b(k) → 0

€

φ(r, t) = e−iω0tψ (r, t)

€

ih∂ψ

∂t= −

h2

2m∇ 2ψThen where

By dropping half of the allowed parameters, Schrödinger reduced the KGE to a 1st order differential equation (in t).

The critical assumption, in detail

€

h2 ∂ 2

∂t 2− c 2h2∇ 2 + m2c 4

⎛

⎝ ⎜

⎞

⎠ ⎟φ(r, t) = 0

The Klein-Gordon Equation

€

ih∂

∂t+

h2

2m∇ 2

⎛

⎝ ⎜

⎞

⎠ ⎟ψ (r, t) = 0

The Schrödinger Eqn. (V=0)

• Halves number of free parameters in the solution.(no longer need and d/dt to solve; just )

• Introduces an explicit time-asymmetry.

• Arbitrary way to halve solutions (Asin+Bcos, A+iB, etc.)

• This particular halving fails in curved spacetime!(Perhaps why QM has never been reconciled with GR)

It’s long past time to revisit the KGE!

The fathers of quantum mechanics never meant to devise a relativistically-correct theory... and yet we’re still using their basic formalism as a starting point 80 years later.

If relativistic thinking was irrelevant, then extrapolating to SR or GR would be simple. The fact that it isn’t easy strongly implies that SR+GR have foundational relevance to QM.

Ambitious Research Goal:

Use KGE to re-derive QM probabilities (associated with preparation-measurement pairs) without dropping b(k).

(i.e. learn how to quantize a second-order differential equation)

The KGE’s “Extra” Free Parameters

The Klein-Gordon equation has a solution with exactly twice the free parameters of the Schrödinger Equation solution But 80 years of experiments say we can’t learn any more information (at one time) than can be encoded by .

(Deeply connected with the Uncertainty Principle)

However, this means one can never get enough information to solve the KGE as an initial boundary condition problem.

Therefore, if we start with the KGE as the master Equation,one gets the axiomatic foundation of Spekkens’s toy model! (We can only know half the total information in .)

Initial Boundary Conditions vs. CPT

Both quantum field theory and relativity are CPT symmetric;should reduce to a time-symmetric non-relativistic picture.

But QM is explicitly time-asymmetric. (The T-asymmetry in the Schrödinger Eqn is fixed “by hand”, the collapse is not.)

To replace the T-asymmetric “collapse” with aCPT-symmetric picture, maybe we shouldn’t be looking

for initial boundary conditions in the first place!

Connection with Huw Price’s work: Asymmetries appear because boundary conditions are imposed asymmetrically.

Boundary conditions are often implemented time-asymmetricallyAtom A emits a photon, and it is later absorbed by atom B:

A B

Using only initial boundary conditions leads to a strange picture:

B

Upon reaching B, the restof the wave “collapses”?!

No time-symmetryin this picture!

A symmetric picturerequires two-time

boundary conditions.

Very symmetric.

CPT and KGE: Natural Partners

A Novel Proposal: Keep the full Klein-Gordon equation. Impose half the boundaries at one time, and half at another time.

Larry Schulman has attempted to impose two-timeboundary conditions on the Schrödinger equation.

Leads to an overconstrainedequation; non-exact solutions.

But the Klein-Gordon equation requires a 2nd boundary condition to determine the “extra” free parameters…

… it can’t go at the beginning, and physical time-symmetry implies it’s far more natural to put it at the end!

“Time’s arrows and quantum measurement”, L.S. Schulman, Cambridge Univ. Press (1997)

Mapping two-time boundaries to QM

If the boundary conditions correspond to measurements, the “collapse” becomes the continuous effect of a future boundary.

Mathematical boundary conditions correspond toexternal physical constraints (i.e. measurements).

(x,t)Time

t=0

t=t0

(Would need both

€

φ(x,0) and∂φ

∂t(x,0).)

Final measurement (procedure + results); allows retrodiction.

Initial measurement (preparation) can’t specify a unique wavefunction.

Two-Boundary FAQs

Doesn’t this violate our intuitive notion of causality?

Yes -- perhaps a benefit in disguise. (Intuition is biased against time-symmetry)

0o45o

45o “+” 0o

Does this permit causal paradoxes?

It’s impossible to retrieve any future-information without changing the boundary conditions.

?

Where does probability fit in?

Huw Price’s pictureof a photon passingthrough 2 polarizers

Discrete Probability WeightsThe 2-boundary problem is solvable, but cannot predict.

Furthermore, once you retrodict the solution , what sense is there to extract an outcome probability from ?

Bayesian answer: “Probability is assigned to propositions, not wavefunctions!”

Fact: Some pairs of boundaries are more likely to occur together than other pairs of boundaries.

If relative weights for each pair are known, one can generate probabilities for any time-biased proposition.

Last semester, did a given student come to class fortwo consecutive lectures?

time

yes

yes

90%

yes

no

5%

no

yes

5%

no

no

0%Student “A”

Student “D”

yes

20%

no

35%

yes

35%

no

10% time

yes yes no no

A Classical Example

Recovered probabilities: If A and D came to previous class, A had a 94.7% attendance probability, while D had 36.4%.

Implementation Questions

This research program comes down to 2 main issues:

• What mathematical boundary condition corresponds toa given physical measurement/constraint?

- Map to existing measurement theory? - Construct GR-friendly measurement theory?

• What is the discrete probability weight thatcorresponds to any complete solution?

- Demand exact correspondence to QM in NR-limit?- Use known results as a guide, not a rule?

Recent Results (arXiv:0706.4075)

Standard theory: Boundary conditions are eigenfunctionsof an operator. (in position space, )

€

ˆ X → x,

€

ˆ P → −ih∇

€

φ(r, t) = a(k)e i(k⋅r−ωt ) + b(k)e i(k⋅r+ωt )dk∫Problem #1: fails for the KGE!

€

ˆ P → −ih∇

€

Q( ˆ X , ˆ P 2)Tentative solution: Use only time-even operators

Propagates in k direction Propagates in -k direction

eigenvalues of both terms are , which does not correspond to physical momentum of the wave

€

ˆ P

€

hk

First Attempt: Two-time Boundary Conditions

€

φ(r, t = 0) = F(r) = F(k)∫ e ik⋅rdkIBC:

€

φ(r, t = t0) = G(r) = G(k)∫ e ik⋅rdkFBC:

t = 0 t = to

Initial BoundaryCondition = F(r)

Final BoundaryCondition = G(r)

€

φ(r, t) = ?

Fourier-expandF(r) and G(r)

Plug into and solve for coefficients

€

φ(r, t)

€

a k( ),b k( ).

€

a(k) =F(k)e iωt0 − G(k)

e iωt0 − e−iωt0

€

b(k) =F(k)e−iωt0 − G(k)

e−iωt0 − e+iωt0

a and b determine ; we know F(k) from initial boundary,G(k) from final boundary, and , but…

€

ω(k) = k 2c 2 + m2c 4 /h2

Next problem: infinite poles

Problem #2; ω is a function of k, so for any value of to, there will always be values of k where ,and the coefficient denominators go to zero!

€

ω t0 = nπ€

a(k) =F(k)e iωt0 − G(k)

e iωt0 − e−iωt0

€

b(k) =F(k)e−iωt0 − G(k)

e−iωt0 − e+iωt0

Import the solution from quantum field theory:give the mass a tiny imaginary component.

Then calculate probability and take limit as

€

ε → 0.

€

h2 ∂ 2

∂t 2− c 2h2∇ 2 + m2c 4 − iε

⎛

⎝ ⎜

⎞

⎠ ⎟φ = 0New KGE:

The “retrodicted” wavefunction

€

φ(r, t) = a(k)e i(k⋅r−ωt )e−εt + b(k)e i(k⋅r+ωt )e+εtdk∫€

a(k) =F(k)e( iω +ε )t0 − G(k)

e( iω +ε )t0 − e−( iω +ε )t0

€

b(k) =F(k)e−(iω +ε )t0 − G(k)

e−(iω +ε )t0 − e+(iω +ε )t0

• No Collapse ( automatically conforms to the final boundary condition)

• Not pre-dictable: need measurement result G(r) (Explains EPR/Bell w/o faster-than-light influences)

In other words, this is a “hidden variable” model that violates Bell’s inequality, because the parameters a(k) and b(k) depend on future events.

Covariant Probability Weight

Charge density of KGE:

€

ρ(r, t) =h

mc 2Im φ

∂φ*

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟

ct

x

t=to

t=0

FBC

IBC

€

W ≡h

mcIm φ

∂φ*

∂η

⎛

⎝ ⎜

⎞

⎠ ⎟

BC

∫Covariant generalization on arbitrary closed boundary:

Here is a unit four-vector, perpendicular to the boundary condition’s 3D hypersurface (inward pointing).

(not well-defined in curved space)

Discrete Probability Postulate

(Wmax-Wmin)2 = P

€

P0(F,G, t0) = F(k)G*(k)e−iωt0 d3k∫2

Known non-relativistic limit:

Given by square of range of W:

€

P ≅ P0 (but not quite!)

Given: 1) Non-relativistic limit 2) Additional time-energy constraint

W has a range because we don’t know the relative phase between F and G, and we don’t know the exact value of to

Four postulates: 3 good, 1 bad• 1) Start with the Klein-Gordon Equation.

(Not the Schrödinger Equation!)

• 2) Constrain with a closed boundary condition in 4-D.(Deal with infinities using m2 => m2-iε)

• 3) Weight the probability with

€

P = Δ Imφ∂φ*

∂ηBC

∫ ⎛

⎝ ⎜

⎞

⎠ ⎟

2

All of these postulates are easily extendible to ageneral relativity framework (curved space), except…

• 4) The boundary condition corresponds to the eigenstate from ordinary quantum measurement theory.

A spacetime view leads to a new perspective of measurements

Standard View:

Preparation

Measurement

time

space

Spatial boundaryconditions

The preparation and spatial boundaries give , from which one calculates the

measurement probabilities.

Spacetime View:

Partial information on ahypersurface constrains thesolution . More solutions lead to a larger weight P.

time

space

hypersurfaceboundarycondition

Physical interactions determine shape and content of boundary conditions

Time

Space

R. Oeckl: “General Boundary Quantum Field Theory”: arXiv.org/hep-th/0509122L. Hardy: “Non-Fixed Causal Structure”: arXiv.org/gr-qc/0608043

Further insight can be found in recent papers:

System

Lab+System

Clues to a GR-friendly measurement theory

• Momentum is not fundamental for fields in GR:

- The stress energy tensor, T, is fundamental.

• On a closed 3-surface (with dual ), one can extract:

- Energy density everywhere on surface: T0

- Momentum density everywhere on surface: Ti

These appear to roughly map to the info in (x,t).

• On a space-like 3-surface, one can integrate the above values to get total energy, angular momentum, etc...

The missing piece of the puzzle...

• Without eigenfunction rule, all possible boundary conditions become reasonable.

(T00(x) need not be localized; is a scalar field)

• Possible paths forward:

- Find probability weight that effectively selects for eigenfunctions.

- New GR-friendly axiom: No paradoxes allowed.

Quantization!

A,B are space-like separated, butcan have a causal effect via

(x,t)

A B

Conclusions• Relativity and CPT symmetry must inform quantum foundations research, even in the non-relativistic limit.

• Both foundations and quantum gravity could benefit from a new interpretation of the Klein-Gordon Equation and a spacetime picture of measurement/boundaries.

This is a hugely ambitious research program...

...but PIAF is the group with the abilities and research inclinations best suited to carrying it out.

Acknowledgements

More information can be found in these papers: K.B. Wharton, “Time-symmetric quantum mechanics”, Foundations

of Physics, v.37 p.159 (2007) K.B. Wharton, “A novel interpretation of the Klein-Gordon

Equation,” arXiv:0706.4075 [quant-ph]

Email: wharton@science.sjsu.edu

Thank you to:- Huw Price, Guido Bacciagaluppi, Centre for Time- Jerry Finkelstein, Lawrence Berkeley Laboratory- Eric Cavalcanti, Griffith University, Australia- Philip Goyal, Perimeter Institute, Canada