multi-time formulation of pair creation - sören petrat, roderich tumulka

8/13/2019 Multi-Time Formulation of Pair Creation - Sren Petrat, Roderich Tumulka

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arXiv:1401.6093v1

[quant-ph]23Jan2014

Multi-Time Formulation of Pair Creation

Soren Petratand Roderich Tumulka

January 23, 2014

Abstract

In a recent work[8], we have described a formulation of a model quantum fieldtheory in terms of a multi-time wave function and proposed a suitable system

of multi-time Schrodinger equations governing the evolution of that wave func-tion. Here, we provide further evidence that multi-time wave functions provide aviable formulation of relevant quantum field theories by describing a multi-timeformulation, analogous to the one in[8], of another model quantum field theory.This model involves three species of particles, say x-particles, anti-x-particles,and y-particles, and postulates that a y-particle can decay into a pair consistingof an x and an anti-x particle, and that an xanti-x pair, when they meet, an-nihilate each other creating a y-particle. (Alternatively, the model can also beinterpreted as representing beta decay.) The wave function is a multi-time versionof a time-dependent state vector in Fock space (or rather, the appropriate productof Fock spaces) in the particle-position representation. We write down multi-time

Schrodinger equations and verify that they are consistent, provided that an evennumber of the three particle species involved are fermionic.

Key words: Multi-time Schrodinger equation; multi-time wave function; rel-ativistic quantum theory; pair creation and annihilation; Dirac equation; modelquantum field theory.

1 Introduction and Overview

This note extends our discussion in [8] of multi-time wave functions in quantum fieldtheory. There, we described a multi-time version of a model quantum field theory (QFT)

involving two particle species,x-particles andy-particles, such thatx-particles can emitand absorb y-particles, x x+ y . Here, we set up a multi-time version of anothermodel QFT involving three species, x-particles,x-particles (which may be thought of asthe anti-particles of the x-particles), andy-particles, with possible reactionsx + x y.

Mathematisches Institut, Ludwig-Maximilians-Universitat, Theresienstr. 39, 80333 Munchen, Ger-many. E-mail: [email protected]

Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: [email protected]

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The model is inspired by electrons (x), positrons (x), and photons (y), but we do not tryhere to be as realistic as possible; rather, we aim at simplicity. In particular, we take allthree species to be Dirac particles and do not exclude wave functions with contributionsof negative energy; also, we ignore any connection between x-states of negative energy

andx-states. We also ignore the fact that our model Hamiltonian is ultraviolet divergent(already in the 1-time formulation) because that problem is orthogonal to the issue ofsetting up a multi-time formulation.

The set of multi-time equations that we propose and discuss in this paper, Equations(12) below, can be applied also to other physical situations besides pair creation. Sincethe equations are independent of whether or notxis the anti-particle ofx, we may writethe reactiony x + xmore abstractly asa b + c. Each of the three species a,b, andccan be chosen to be either fermions or bosons, and assigned an arbitrary mass. Thus,besides pair creation, another scenario that is included is, as a variation of the reactionx x+y of [8], the reaction x z+y in which an x-particle emits a y-particle andmutates into az-particle; conversely, azabsorbing ay becomes anx. For example, beta

decay is of this type; in fact, beta decay (roughly speaking, the decay of a neutron intoa proton, an electron, and an anti-neutrino) actually consists of two decay events, ofwhich the first is du+W (a down quark mutates into an up quark while emittinga negatively charged Wboson), and the second is W e +e (the W decays intoan electron and an anti-e, where e is an electron-neutrino); that is, each of the twodecay events is of the form xz+y.

We remark that all fundamental processes of particle creation or annihilation innature seem to be of one of the three forms x x+ y , x z+ y, or y x+ x.Our work in [8] and in the present paper shows that all of these processes can beformulated in terms of multi-time equations. Thereby, it contributes evidence in favor

of the hypothesis that multi-time wave functions provide a viable formulation of allrelevant QFTs.Multi-time wave functions were considered early on in the history of quantum theory,

also for quantum field theory [2, 3, 1]. However, in these early papers, only electronswere regarded as particles, and photons were replaced by a field configuration. Bloch [1]noted that multi-time equations require consistency conditions (see also[7]). Multi-timewave functions with a variable number of time variables (see below for explanation) wereconsidered before in [10, 4, 5, 6, 8]. Droz-Vincent [4, 5] gave an example of a consis-tent multi-time evolution with interaction which, however, does not correspond to anyordinary QFT model. For further references about multi-time wave functions, a deeperdiscussion of consistency conditions, and no-go results about interaction potentials in

multi-time equations, see [7]. For a comparison of the status and significance of multi-time formulations in classical and quantum physics, see [9].

The plan of this paper is as follows. After introducing notation in Section2, we firstdescribe the model QFT in Section3in the usual one-time formulation and then givea multi-time formulation in Section4; in particular, we specify a system of multi-timeSchrodinger equations1 for the multi-time wave function , see (12). The function

1The expression Schrodinger equation is not meant to imply that the Hamiltonian involves the

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is defined on the set Sxxy ofspacelike configurations, which means any finite numberof space-time points that are mutually spacelike or equal,2 with each point markedas either an x-, an x-, or a y-particle. The set Sxxy can be regarded as a subset of(R4) (R4) (R4), with one factor for each particle species, R4 representing space-time, and the notation

(S) =

N=0

SN (1)

for any setS. Since a functionfon (S) can be thought of as consisting of one functionf(N) on each sector SN, and since f(N) is a function ofNvariables with values in S,we say that f is a function with a variable number of variables. Correspondingly, on Sxxy involves a variable number of time variables. If all time variables in are setequal (relative to some Lorentz frame L), we recover the wave function of the 1-timeformulation. We then verify the consistency of our proposed system of equations inSection5; that is, we verify (on a non-rigorous level) that the system possesses a unique

solution for every initial datum 0 (at all times set equal to zero). Our proposedmulti-time QFT model is not fully covariant because it involves, as a coefficient of thepair creation term, an arbitrary but fixed complex-linear mapping g : S SS,with S = C4 the 4-dimensional complex spin space used in the Dirac equation, andnonzero mappings g of this type are never Lorentz-invariant. Apart from this point, themodel is covariant, as can be seen from the Lorentz transformation of the multi-timeSchrodinger equations that we carry out in Section 6. The natural generalization tocurved space-time is described in Section7.

As a last remark in this section, we find that the multi-time equations of the reactiona b+ c are consistent if and only if an evennumber of the three species a,b,c arefermions, and an odd number of them are bosons (also in case some of the speciescoincide, such as a = b). That is, either all three are bosons, or two of them arefermions and one is a boson. This rule for which kinds of reactions are allowed agreeswith which kinds of reactions occur in nature.3 This rule can also be derived fromthe conservation of spin and the spinstatistics relation (i.e., that bosons have integerand fermions half-odd spin), but here we obtain it differently, from the consistency ofthe multi-time equations. That is, a single-time formulation (i.e., a Hilbert space andHamiltonian) can be (consistently) specified for each of the 8 combinations of a,b,cbeing bosons or fermions, but in the multi-time formulation the rule above restricts thepossibilities, thereby providing a possible explanation of why reactions violating the ruledo not occur in nature.

Laplace operator, but is understood as including the Dirac equation.2Note our usage of the word spacelike: Two space-time points that are mutually spacelike are

unequal, but a spacelike configuration may contain two particles at the same space-time point.3Indeed, a fermion cannot decay into two fermions, nor into two bosons, whereas a boson can decay

into two fermions (e.g., photon electron + positron) or two bosons (e.g., Higgs W+ + W);furthermore, a fermion can decay into a fermion and a boson (e.g., du + W) but a boson cannot.

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Equivalently, the Hamiltonian can be written as

H=Hx+Hx+Hy+Hint (9)

with

Hx=

d3x

4r,r=1

ar(x)irr +mxrrar(x) (10a)

Hint=

d3x

4r,r,s=1

grrsa

r(x) a

r(x) bs(x) +g

rrsar(x) ar(x) b

s(x)

(10b)

and expressions analogous to (10a) for Hx and Hy. Here, denotes the adjoint oper-

ator, and as(x), as(x), bs(x) the annihilation operators for an x, x, y-particle with spincomponents at location x in position space, explicitly defined byar(x) (x3M, x3M, y3N) = M+ 1Mx rM+1=r(x3M,x), x3M, y3N , (11a)

ar(x)

(x3M, x3M, y3N) = 1

M

Mj=1

j+1x rrj3(xjx) rj

x3M \ xj, x3M, y3N

(11b)

and correspondingly foras(x) and bs(x).

4 Multi-Time Formulation

The constant R that we use below can be chosen to be 1/2; we will show inRemark3 below that its value actually does not matter. The multi-time wave function is supposed to be defined on Sxxy and have values (x

4M, x4M, y4N) S(M+M+N)

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2. As discussed in Remark 5 in Section 2.2 of[8], something needs to be said abouthow (12) should be understood at the tips ofSxxy (i.e., at configurations with twoparticles at the same location, henceforth called collision configurations). That isbecause, for example at a configuration with xj =yk,/x

0j does not make sense

for a function on Sxxy as varying x0

j while keeping yk fixed will lead away fromSxxy. The rule formulated in [8] for these cases should be adopted here as well:At such a configuration, (12) should be understood as prescribing the directionalderivative normally written as (/x0j +/y

0k) (and correspondingly for other

collisions).

3. The system (12) for any value of R is actually equivalent to the system (12) forany other value of R. Indeed, it is known [11, Thm. 1.2 on p. 15] that the Greenfunction of the Dirac equation (such as G and G) vanishes on spacelike vectors;thus, the terms involving vanish at collision-free spacelike configurations. Theydo not vanish at a collision between (say) xj andx; there, according to Remark2

above, the equations (12) are understood as specifying not /x0j and /x

0

individually but only their sum, from which the factor cancels out.

Thus, the solutionof (12) is independent of; put differently, the one multi-timeevolution law possesses several representations by systems of equations (involvingdifferent values of), roughly analogous to the way a gauge field possesses severalrepresentations corresponding to different gauges. That is, the choice of is amatter of mere aesthetics; natural choices are = 0, = 1/2, or = 1.

4. In the same way as for Assertion 3 in [8], one can show that the multi-time wavefunction of (12) can be expressed as follows in terms of the particle annihilation

operators at any (x4M, x4M, y4N)Sxxy:

x4M, x4M, y4N

=

M(M1)/2x

M(M1)/2x

N(N1)/2y

M!M!N!

ar1(x1) arM(xM) ar1(x1) arM(xM) bs1(y1) bsN(yN)0 . (15)

Here, 0 is obtained from by setting all time variables equal to zero, andar(t,x) =e

iHtar(x)eiHt is the Heisenberg-evolved annihilation operator with the

single-time HamiltonianHas in (7b).

5. For any spacelike hypersurface , let be the restriction of to ()3 (i.e.,

to configurations on ), and let = F0 be the interaction picture wavefunction, where 0 is the hypersurface of t = 0 (in the Lorentz frame L) andF12 is the free time evolution from 1 to 2. Then

obeys the TomonagaSchwinger equation

i

=

d4x HI(x)

(16)

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for infinitesimally neighboring spacelike hypersurfaces , , with the interactionHamiltonian density in the interaction picture given by

HI(t,x) =e

iHfreet4

r,r,s=1grrsar(x) ar(x) bs(x) +grrsar(x) ar(x) bs(x)eiHfreet .

(17)This can be derived in much the same way as Assertion 5 in [8], using Assertion 10of [8].

5 Consistency

As discussed in detail in [7], there is a non-trivial condition for a system of multi-timeequations to be consistent, i.e., to possess a solution for arbitrary initial conditions attime 0 (i.e., all times equal to 0). In Sections 5.15.5 of [8], we have shown for multi-time

equations of the type considered there that (leaving aside the ultraviolet divergence) theyare consistent on the set of spacelike configurations if, at every spacelike configurationq, the consistency condition

i

tjHj , i

tkHk

= 0 (18)

holds for any two non-collidingparticles j, k belonging to q (see in particular Footnote13 in Section 5.5 of[8]). The arguments described there apply to (12) as well. Therefore,the consistency of (12) follows if (18) holds for any two non-colliding particles, whereHjis the right-hand side of (12a), (12b), or (12c), whichever is appropriate depending

on the species of particle j .This is indeed the case if (14) holds. We report here the commutators. Any pair of

particles is of one of these six forms: xx,xx, yy,xx, xy, or xy . Thus, six commutatorsneed to be computed. Let us begin with the yy commutator (and note that N 2whenever there are two y variables for which we can consider the commutator):

iy0kHyk , iy0 Hy

=

(M+ 1)(M+ 2)(M+ 1)(M+ 2)

N(N1)4

rM+1,rM+2,rM+1,rM+2=1

grM+1rM+1skgrM+2rM+2s

k+y (xxy1)rM+1rM+2rM+1rM+2sk s(x4M, yk, y), (x4M, yk, y), y4N \ {yk, y},(19)

assumingk < . This is a rather unwieldy expression, but the only aspect that mattersat this point is the factor xxy1, which makes the whole expression vanish if (14)holds. Here are all six commutators, with those terms containing a factor ofxxy1

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with implicit summation over but not over j or or k , and

Gs

rr (x) =4

r=1(0)rrGrrs(x) (22a)

G srr (x) =4

r=1

(0)rrGrrs(x) (22b)

(g+)rrs=4

s=1

(0)ssgrrs. (22c)

In (21) and the left-hand side of (22), upper spin indices refer to S, the dual space ofS, while lower ones refer to S. The Lorentz-invariant operation + is defined in Section2.4 of[8].

The system of equations (21) is Lorentz-invariant if we regard G, G as functions

M SSS

, where Mdenotes Minkowski space-time, and g+ as an element ofS S S.

In fact, Gs

rr (x) is the Green function (in the variable x and the spin index r) withinitial spinor covariantly characterized by g. Likewise, G srr (x) is the Green function(in the variable xand the spin index r) with initial spinor covariantly characterized by

g. That is, the objects G, G and g+ can all be obtained in a unique and covariant way(and with the right transformation behavior) once an element gSS S has beenchosen.

7 Generalization to Curved Space-Time

In Section 2.5 of [8], we had outlined the rather straightforward way how the multi-timeequations presented there can be generalized to a curved space-time (M, g). The multi-time system (12) or, equivalently, (21) can be adapted to curved space-time in the sameway. Here is a brief overview. In this setting, is defined on the spacelike configurationsin (M)3 with values

(x1, . . . , xM, x1, . . . , xM, y1, . . . , yN)Sx1 SxMSx1 SxM Sy1 SyN, (23)

whereSxis a fiber of the bundleSof spin spaces, a vector bundle over the base manifoldM. This can be expressed using the notation AB for the vector bundle over the basemanifold A B (obtained from vector bundles A over A and B over B) whose fiberat (a, b) AB is

(A B)(a,b) = AaBb, (24)and correspondingly An for A A A with n factors. Then the (M , M , N )-particle sector of is a cross-section of the vector bundle S(M,M,N) =SMSMSN

over MM+M+N. The equations (21) need only minor changes and re-interpretation of

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symbols, such as that j is now the covariant derivative onScorresponding to the con-

nection naturally associated with the metric ofM, and correspondingly onS(M+M+N).The coupling matrix g srr gets replaced in (21c) by a cross-section g

srr (yk) of the bundle

S

S

S. Similarly, G(x

x) in (21a) gets replaced by G(x, x), which is the appro-

priate Green function, namely the solution of the free Dirac equation in x with initialspinor at x covariantly characterized by g(x); likewise, G(xx) in (21b) gets replacedby the appropriate Green function G(x, x).

Our consistency proof still applies.

Acknowledgments. S.P. acknowledges support from Cusanuswerk, from the GermanAmerican Fulbright Commission, and from the European Cooperation in Science andTechnology (COST action MP1006). R.T. acknowledges support from the John Temple-ton Foundation (grant no. 37433) and from the Trustees Research Fellowship Programat Rutgers.

References

[1] F. Bloch: Die physikalische Bedeutung mehrerer Zeiten in der Quantenelektrody-namik. Physikalische Zeitschrift der Sowjetunion, 5:301305 (1934)

[2] P. A. M. Dirac: Relativistic Quantum Mechanics. Proceedings of the Royal SocietyLondon A, 136:453464 (1932)

[3] P. A. M. Dirac, V. A. Fock, and B. Podolsky: On Quantum Electrodynamics.Physikalische Zeitschrift der Sowjetunion, 2(6):468479 (1932). Reprinted in J.

Schwinger: Selected Papers on Quantum Electrodynamics, New York: Dover (1958)[4] Ph. Droz-Vincent: Second quantization of directly interacting particles. Pages 81

101 in J. Llosa (ed.): Relativistic Action at a Distance: Classical and QuantumAspects, Berlin: Springer-Verlag (1982)

[5] Ph. Droz-Vincent: Relativistic quantum mechanics with non conserved number ofparticles.Journal of Geometry and Physics, 2(1):101119 (1985)

[6] H. Nikolic: QFT as pilot-wave theory of particle creation and destruc-tion. International Journal of Modern Physics A, 25:14771505 (2010)http://arxiv.org/abs/0904.2287

[7] S. Petrat and R. Tumulka: Multi-Time Schrodinger Equations Cannot ContainInteraction Potentials. Preprint (2013) http://arxiv.org/abs/1308.1065

[8] S. Petrat and R. Tumulka: Multi-Time Wave Functions for Quantum Field Theory.Preprint (2013) http://arxiv.org/abs/1309.0802

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[9] S. Petrat and R. Tumulka: Multi-Time Equations, Classical andQuantum. To appear in Proceedings of the Royal Society A (2014)http://arxiv.org/abs/1309.1103

[10] S. Schweber: An Introduction To Relativistic Quantum Field Theory. Row, Petersonand Company (1961)

[11] B. Thaller: The Dirac Equation. Berlin: Springer-Verlag (1992)

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multi-time formulation of pair creation - sören petrat, roderich tumulka

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