Path integration in relativistic quantum mechanics

arXiv:gr-qc/9210019v1 28 Oct 1992Path integration in relativistic quantum mechanicsIan H. Redmount and Wai-Mo SuenMcDonnell Center for the Space SciencesWashington University, Department of PhysicsSt. Louis, Missouri63130–4899USATo be published in International Journal of Modern Physics A.The simple physics of a free particle reveals important features of the path-integral formulation of relativistic quantum theories.The exact quantum-mechanical propagator is calculated here for a particle described by the simplerelativistic action proportional to its proper time.

This propagator is nonva-nishing outside the light cone, implying that spacelike trajectories must beincluded in the path integral. The propagator matches the WKB approxima-tion to the corresponding configuration-space path integral far from the lightcone; outside the light cone that approximation consists of the contributionfrom a single spacelike geodesic.

This propagator also has the unusual prop-erty that its short-time limit does not coincide with the WKB approximation,making the construction of a concrete skeletonized version of the path integralmore complicated than in nonrelativistic theory.1

1. IntroductionThe path-integral formulation of relativistic quantum mechanics gives rise toproblems not found in nonrelativistic theory.

Some of these are similar to problemswhich arise in attempting to construct a quantum description of gravity. In thiswork we seek to elucidate some of these problems by considering the much simplerphysics of a single, free, relativistic particle, a system offering the advantage of exactsolubility.The quantum mechanics of a particle can be completely described by a prop-agator, given by a functional integralG(x, t; x0, t0) =ZC Dx eiS/¯h .

(1.1)Here C denotes the class of paths included in the integral, and S is the classicalaction associated with each path. In nonrelativistic theory, C properly includes allpaths linking spacetime point (x, t) to (x0, t0) which move forward in time, in thesense that such a class C in (1.1) gives the same propagator as that in the canonicalHamiltonian quantization.

In a relativistic theory, with Lorentz-invariant action S,the light-cone structure of the spacetime comes into play: Should C include all pathsthat move forward in time, or only those inside the light cone, i.e., paths which arealways timelike? If the latter, then the propagator (1.1) must vanish outside thelight cone of (x0, t0).

If spacelike paths are included, this will not be so in general,and G will describe propagation outside the light cone.Such propagation will be acausal—backward in time in some Lorentz frames.Whether such a feature is admissible, or perhaps necessary, is a question which2

arises in quantum gravity as well as in particle mechanics. Indeed, Teitelboim [1]has argued that a quantum gravity theory cannot be both covariant and causal; hesuggests [2] retaining causality, although Hartle [3] argues that covariance shouldbe preserved instead.

There too the problem can be framed as a choice of historiesto be included in a path integral: It is the choice between the class of spacetimehistories which includes those with negative lapse function, i.e., backward timedisplacement, and that which does not [2,3]. (In fact the action for general relativitycan be converted into a form [4] like that for a relativistic particle; general relativityis analogous to parametrized theories for such particles [2,3,5].

)Here we evaluate the quantum-mechanical propagator for a free, relativisticpoint particle. We show the significance of spacelike paths by comparing the exactpropagator with the WKB approximation to the formal, non-Gaussian, configuration-space path integral.Comparison of the exact and WKB expressions for the propagator in the short-time-interval limit also yields the appropriate measure for the path integral.

Withthis a concrete “skeletonized” version of the formal integral can be constructed. Heretoo the relativistic theory encounters complications not found in the nonrelativisticcase.For simplicity we treat a particle in 1 + 1-dimensional flat spacetime; gener-alization to higher dimensions is straightforward.

Units with ¯h = c = 1 are usedhenceforth.3

2. Propagator and path integral for a free relativistic particleA single free particle can be described by the relativistic actionS = −Zm dτ = −Zm(1 −˙x2)1/2 dt ,(2.1)where m is the particle’s mass, τ its proper time, and (x, t) its coordinates insome Lorentz frame, with ˙x ≡dx/dt.Hence this model has Lagrangian L =−m(1 −˙x2)1/2, momentum p = ∂L/∂˙x = m˙x(1 −˙x2)−1/2, and HamiltonianH = p˙x −L = +(p2 + m2)1/2 .

(2.2)Being nonpolynomial in p, this corresponds to a nonlocal quantum operator; it isto be interpreted as acting on any wave functionψ(x, t) =Zdk eikx φ(k, t)(2.3a)to give [6,7]Hψ(x, t) =Zdk eikx (k2 + m2)1/2φ(k, t) . (2.3b)The sign of H in Eq.

(2.2) emerges unambiguously from the canonical formalism,given the action (2.1). It implies that the quantum-mechanical description is en-tirely in terms of positive-frequency functions.

Such a description is adequate for anoninteracting particle [6]. [It is formally the same as the positive-frequency branchof a Klein-Gordon theory.

In the Klein-Gordon picture interactions could “scatterthe particle into the negative-frequency branch.” Teitelboim [2], however, has con-structed a theory of interacting relativistic point particles based on an action ofform (2.1), plus interaction terms. ]4

The propagator for the wave equation i ˙ψ = Hψ is given by the integralG(x, t; x0, t0) =Z +∞−∞dk2πeik∆x e−i(k2+m2)1/2∆t ,(2.4)with ∆x ≡x −x0 and ∆t ≡t −t0; clearly this is a solution of the equation, with“initial value” δ(∆x) at ∆t = 0. The same integral expression can be obtained usingthe localized states of Newton and Wigner [7], i.e., as the projection of the statelocalized at x0 at time t0 on that localized at x at time t. Hartle and Kuchar [5]derive this same “Newton-Wigner propagator” via a phase-space path integral.A very simple calculation yields the propagator G in closed form: The inte-gral (2.4) can be evaluated by adding a small negative imaginary part to ∆t forconvergence.

The result isG = limǫ→0+m(i∆t + ǫ)πλ1/2ǫK1(mλ1/2ǫ) ,(2.5)with λǫ ≡(∆x)2 + (i∆t + ǫ)2 and K1 the familiar modified Bessel function. Thispropagator contains the complete description of the behavior of a free particle inthis formulation of relativistic quantum mechanics.The corresponding configuration-space path integral (1.1) for the propagatoris, formally,G =ZDx exp"−imZ (x,t)(x0,t0)(1 −˙x2)1/2 dt#.

(2.6)The treatment of spacelike paths in this functional integral is clearly problematic [2].If such paths are to be excluded from the integral by fiat then G must vanish forall (x, t) outside the light cone of (x0, t0), i.e., for |∆x| > |∆t|. But expression (2.5)does not do this: Outside the light cone mλ1/2ǫis (nearly) real, and the Bessel5

function is nonvanishing—decreasing exponentially for large argument. Obviouslythe path integral for a relativistic particle must include the contributions of spacelikepaths in general.Such contributions are manifest in the WKB approximation to the propagator.In such circumstances as the integral (2.6) is dominated by paths near the classicaltrajectory between (x0, t0) and (x, t), it is approximated by the WKB expression [8]ZDx eiS ∼ i2π∂2Scl∂x∂x0!1/2eiScl ,(2.7)with Scl the action evaluated along that classical path.

Here that path simply hasconstant speed ˙x = ∆x/∆t. Hence the classical action is Scl = +imλ1/2ǫ(ǫ →0+),and the resulting approximation isG ∼limǫ→0+ m(i∆t + ǫ)22πλ3/2ǫ!1/2e−mλ1/2ǫ.

(2.8)(Here the ǫ term serves to specify the phases of the square roots.) The exact propa-gator (2.5) approaches just this form in the regime m|λ1/2ǫ| ≫1, i.e., many Comptonwavelengths from the light cone; in other words, as expected, the WKB approxima-tion to the path integral (2.6) is accurate when the magnitude of the classical actionis large compared to ¯h.

For (x, t) well outside the light cone of (x0, t0), the dominanttrajectory giving rise to form (2.8) of the propagator is the spacelike geodesic (line)between the two points.These features of the propagator are illustrated in Fig. 1, which shows theevolution of a simple Gaussian initial wave function via G. The extension of the6

propagator outside the light cone and the coincidence of the exact and WKB formsaway from the light cone are evident.Unlike familiar nonrelativistic propagators, the relativistic-particle propagatordoes not coincide with the WKB form in the limit ∆t →0. In the regime |∆x|, |∆t| ≪m−1, the exact propagator (2.5) takes the formG(exact) ∼1π limǫ→0+i∆t + ǫλǫn1 + O[m2|λǫ ln(mλ1/2ǫ)|]o(2.9a)while the WKB approximation (2.8) takes the formG(WKB) ∼ m2π1/2limǫ→0+i∆t + ǫλ3/4ǫh1 + O(m|λ1/2ǫ|)i.

(2.9b)At ∆t = 0 the former becomes δ(∆x), as required; the latter does not. In this dis-agreement between the ∆t →0 and WKB limits, the relativistic-particle propagatorresembles certain curved-space propagators [9].The absence of a factor eiScl = 1−mλ1/2ǫ+O(m2|λǫ|) in form (2.9a) might leadone to conclude that a Lagrangian, i.e., configuration-space, path integral for therelativistic particle cannot be constructed, as indicated by Hartle and Kuchar [5].This can be done, however, by including appropriate factors in the path-integralmeasure.

Thus a skeletonized version of integral (2.6) can be given as the compo-sitionG(x, t; x0, t0) = limN→∞Z +∞−∞dx1 · · ·Z +∞−∞dxN G(x, t; xN, tN) · · ·G(x1, t1; x0, t0) ,(2.10a)with the infinitesimal-interval propagatorsG(xk, tk; xk−1, tk−1) = limǫ→0+"m(i∆t + ǫ)πλ1/2ǫK1(mλ1/2ǫ) exp(mλ1/2ǫ)#eiScl ,(2.10b)7

where ∆t, λǫ, and Scl are taken between (xk−1, tk−1) and (xk, tk). This form followsdirectly from the exact result (2.5); since ∆x, hence λǫ, need not be small comparedto m−1 even when ∆t is, no expansion of that result is suitable.The measurefactors, those in square brackets preceding eiScl in Eq.

(2.10b), are considerably morecomplicated than their nonrelativistic counterparts. This is to be expected since thekinetic term in the relativistic action is more complicated than the nonrelativisticterm, which is simply quadratic in the velocity.3.

ConclusionsThe free, relativistic point particle provides a simple, exactly soluble exampleof some remarkable features of path integrals for relativistic quantum theories. TheNewton-Wigner [7] propagator for such a particle can be expressed in the closedform (2.5).Although the corresponding configuration-space path integral (2.6)cannot be evaluated exactly [except in the sense that Eq.

(2.5) is the evaluation ofintegral (2.6)], it can be approximated by the formal WKB expression (2.8). TheWKB approximation can be calculated without knowing the precise form of thepath integral measure.

It agrees with the exact result far from the light cone, whichsubstantiates the suitability of the path-integral formulation of this theory. Theexact and approximate forms are seen to differ only near the light cone, i.e., wherethe classical action is not large compared to ¯h, as expected.The propagator is manifestly nonvanishing outside the light cone.

Hence thepath integral must include the contributions of spacelike trajectories; in particular8

spacelike classical paths give the dominant contribution in the construction of theWKB approximation far outside the light cone, where that approximation is valid.The exact and WKB forms of the propagator show a difference between theshort-time and WKB limits, i.e., the limits ∆t →0 and ¯h →0, of the relativistic-particle propagator, unlike its nonrelativistic counterpart. This discrepancy intro-duces additional complexity into the relativistic theory, i.e., into the path-integralmeasure.

This does not appear to invalidate the configuration-space path-integralformulation [5], though it does diminish its intuitive appeal.The contribution of spacelike paths to the relativistic-particle path integral,i.e., the fact that the propagator is nonvanishing outside the light cone, shows thatacausality is a feature of at least this simple relativistic quantum theory. Strictcausality is recovered as a classical limit: The propagator falls offexponentiallyoutside the light cone, on a scale of the particle’s Compton wavelength.

It might beargued that a first-quantized theory is inappropriate, that quantum field theory iscalled for. Nonetheless the same questions of acausality and the proper domain ofhistories in the path integral are known to arise in quantum gravity [1–3,10].

Theexample treated here lends support to the conjecture [3] that path integrals in thatmore complex case should also include the contributions of acausal histories.AcknowledgementsWe thank C. Bender, M. Visser, C. M. Will, and K. Young for useful dis-cussions in the course of this work. Financial support was provided by the U. S.9

National Science Foundation via Grants No. PHY89–06286 and No.

PHY85–13953.References[1] C. Teitelboim, Phys. Rev.

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[8] L. S. Schulman, Techniques and Applications of Path Integration, (Wiley, NewYork, 1981), p. 94.

[9] Schulman, Techniques and Applications of Path Integration (Ref. 8), pp.

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Figure CaptionFIG. 1.

Real (a) and imaginary (b) parts of a wave function propagated via therelativistic-particle propagator G. The wave function with initial form ψ(x, 0) =exp(−m2x2) is shown at time t = 10m−1. The solid curves are the exact wavefunction, the dotted that obtained using the WKB approximation to the propagator.The values of ψ are in units of m.11

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