The Feynman Propagator from a Single Path G. N. Ord∗ M.P.C.S. Ryerson University Toronto Ont. J. A. Gualtieri† Applied Information Sciences Branch Global Science and Technology Code 935 NASA/Goddard Space Flight Center Greenbelt MD. (Dated: September19 2001) 2 We show that it is possible to construct the Feynman Propagator for a free particle in one 0 dimension, without quantization, from a single continuous space-time path. 0 2 n The Feynman path-integral formulation of Quantum 800 a Mechanics[1, 2] is well known for its utility and intuitive J 700 appeal. An interesting history of its development may 9 be found in the article and book by Schweber[3, 4]. Al- 600 2 though the mathematics of the path integral encourages 2 ustothinkofthepathsintermsofrealspace-timetrajec- 500 v tories,andtherehavebeenveryinterestingproposalsfor 400 2 testing the reality of the paths [5, 6, 7], the formulation 9 itself falls short of providing a full microscopic basis for 300 0 9 quantum mechanics. This is in contrast to the Wiener 200 integralwhichisanabstractionofthemicroscopicmodel 0 1 (Brownianmotion)supportingthediffusionequation. In 100 0 particular, Wiener paths are known to approximate ac- 0 h/ tual physical trajectories of diffusing particles, whereas -100 -50 0 50 100 150 200 p the relation between Feynman paths and physical parti- FIG.1: AFeynmanChessboardtrajectory. Thex-axisishor- - cles is not so direct. izontal and the t-axis vertical. The sign of the contribution t n Therearetwomainbarriersto anassociationbetween changeseverytwocornersinthetrajectory. Thisisindicated a Feynmanpathsandanyphysicaltrajectoryofarealpar- in the figure by the different line widths in the different seg- u ments. ticle. First of all there is a many-to-one correspondence q : betweenFeynmanpathsandtheparticlebeingdescribed. v Interference effects require this non-uniqueness since in- i function without invoking an analytic continuation. The X dividualtrajectoriescarryvariablephasebutnotvariable propagatorappearsnaturallyasapatterncreatedbythe r amplitude in the propagator [8]. Thus a physical parti- (space-time) plane-filling path of a single point-particle. a cle cannot simply traverse a single Feynman path while Inthe new formulation,the many-to-oneaspectofFeyn- propagating in space-time. manpathsiscircumventedbysewingtogetheranensem- A second impediment is that, in the path integral for- bleofChessboardpathsintoasinglecurveinsuchaway mulation, the required reduction of wave functions on that formal quantization is unnecessary. measurement is grafted onto the dynamics of propaga- The Chessboard or Checkerboard model[2, 9, 10] ex- tion; itdoesnotfollowinadirectfashionfromthe paths tendedFeynman’spathintegralapproachtotherelativis- themselves. As in other formulations of quantum me- tic domain in order to incorporate electron spin. In this chanicsweneedmeasurementpostulatestointerpretthe model, particles hop with speed ±c on a discrete space- theory in terms of the real world. time lattice with spacing ǫ. Choosing units in which In this paper we show that in the particular case of c=1, paths consistofdiagonalsegmentsresemblingfor- the Feynman Chessboard model, one can modify the ward bishop’s moves in chess(Fig. 1). formulation so that the propagator can be constructed AlatticeapproximationtotheKernelK(b,a)forapar- by a single continuous space-time curve. This is done ticle to propagate from position a at time t to position a by allowing particles to have trajectories with reversed b at time t is given by Feynman to be: b time segments. Although this might seem conceptually ‘expensive’, allowing this feature explicitly provides the K(b,a)= N(R)(iǫm)R (1) physical mechanism which creates the phase of a wave X R 2 where the sum is over all Chessboard paths and N(R) 800 is the number of paths with R corners. Here m is the 700 mass of the particle in units where h¯ is one. In terms of the paths themselves, the expected distance between 600 corners is 1/m [10]. If we distinguish between the two 500 directions in space, K is a 2x2 matrix which converges to the Dirac propagator in the continuum limit[9]. The 400 prescription given in (1) can be modified somewhat for 300 convenience. Gersch, who established the relation be- tween the Chessboard model and the one dimensional 200 Ising model, pointed out that the non-relativistic limit is more direct if i is replaced by −i in (1). Kull and 100 Treumann[11] also noted that paths fixed at both ends 0 have (R−1) degrees of freedom, so the R in (1) may be -100 -50 0 50 100 150 200 replaced by (R−1) without interfering with the contin- FIG. 2: The Chessboard Trajectory of Fig. 1 and its Or- uum limit. thogonal Twin. This pair can be viewed as two osculating Equation (1) is a formal analytic continuation (quan- Chessboard paths which never cross, or as a single entwined tization) of a classical partition function. The i in the loop which crosses itself frequently. The latter view explains sum, which replaces a real positive weight in the parti- the phase shift of π for every two corners in the Chessboard paths. tionfunction,enforcesthequantization. Italsopartitions the sum into 4 components, each of which is real, i.e.: K(b,a) = ( N(R)(ǫm)R− N(R)(ǫm)R) behavelikeantiparticlesinthattheyreducethecontribu- X X tion of the particles, providing interference effects. The R=0,4,... R=2,6,... ensembleofsuchcolouredpathsbetweenaandbprovides + i( N(R)(ǫm)R− N(R)(ǫm)R) X X the appropriate contribution to a quantum propagator, R=1,5... R=3,7,... but is not explicitly traversedas a single path. What we = Φ +iΦ . (2) R I wouldlike to dois to sewtogetherthe Chessboardpaths insuchawaythattheymaybetraversedbyasinglepath Each of the above sums is, by itself, a partition func- whichalso providesthe alternatingcolours ofthe trajec- tionforaclassofrandomwalksinwhichtheterm(ǫm)R tories through the direction in time of the traversal. To is just a Boltzman weight. The interference of alterna- thisend, wenotefromFig. 2thateachChessboardpath tive paths is a result of the two subtractions in (2). If has an orthogonaltwin. we replace the minus signs in (2) by plus signs, the re- The orthogonal twin starts from the origin moving in sulting propagator is related to the Telegraph equation, theoppositedirectionwiththeoppositecolour. Itmoves which in turn becomes the diffusion equation in the ap- thesamedistanceasthesecondlegofitstwin’spath,re- propriate‘non-relativistic’limit[12],theremainingithen versesdirectionandmovesthesamedistanceasitstwin’s being superfluous. The underlying stochastic model for first leg. Twins meet at every second corner where they this case has been studied by Kac[13] and its relation changebothcolouranddirection. Forpathswithanodd to the Dirac equation through analytic continuation has number of corners, this is repeated until the twins meet been discussed by Gaveau et.al.[14] and Jacobson and at t = t (for paths with an even number of corners see Schulman[10]. Withtheoriginalminussignsinplace,the b below). The orthogonal twin is also a Chessboard path i which appears in (2) just expresses K as a particu- with colouring 180o out of phase with the original. larlyconvenientlinearcombinationoftherealamplitudes Φ , however the actual interference characteristic of Now consider the following ‘entwined’ traversal of the R/I quantization is apparent in the oscillatory nature of the two paths. Follow the first twin to the first meeting, the Φ themselves. secondtothesecondmeetingandsoon. Thispathisblue Since it is the occurrence of the minus signs in the from the origin to the last meeting. From there reverse propagatorwhich is essentialfor interference we look for the direction in t by proceeding down the remaining red aphysicalbasisforthesubtractions. RegardingFig.1we sections. Thisbringsyoubacktotheoriginonanentirely canencodethecountingandsubtractionsinvolvedin(2) redpath. This choiceof traversalgivesa meaning to the by colouring the trajectories with two colours, say blue originalFeynmancolouring;thecolouringcorrespondsto (thick lines in figure) and red(thin in figure). If the tra- the directionin time ofanentwined path traversal. Blue jectoriesstartoutblue, they changeto redatthe second correspondstoforwardint,redtobackwards. Entwined corner,blueatthefourthandsoon. Thesignofthecon- pairsalsoconservechargeifweassociateoppositecharges tribution ofa trajectoryis then determined by its colour with reversed time segments. attheendpoint,+forblue,–forred. Redcontributions Eachchessboardtrajectoryin(2)hasauniqueorthog- 3 onal twin. Let P be an arbitrary n-step R-cornered R Chessboard path. Write PR = (σ1,σ2,...,σn) where σ =±1accordingtothedirectionofthek-thstepofthe k path. Ifwedefinea‘leg’asasetofcontiguousstepsallin thesamedirectionandboundedbyeithercornersorends ofapath(i.e. adomainintheIsinganalogy),thenifRis odd,wemaywritePR =(l1,l2,...,lR+1)withtheunder- standing that l1 stands for the first leg, l2 stands for the second and so on. If R is even then the path ends with the lastlink inthe samedirectionasthe firstlink. Inor- dertojointhepathtoanorthogonaltwinweneedtoadd a final leg in the opposite direction. To do this uniquely we add a final leg the same length as the original last leg but in the opposite direction. Thus if R is even, we FIG.3: Thesumofpropagatorcomponents,ΦR+ ΦI along extend the n-step path to PR =(l1,l2,...,lR+1,−lR+1), thex-axis,fromtheChessboardmodel(curve)andthesingle where −lR+1 is lR+1 with the signs of all the component path simulation (points) at t=15 stepsfrom theorigin. σ changed. We may then define the orthogonal twin to P as R uniform coverage of the ensemble. (−l1,l1) R =0 If we allow a walker to cycle through the entwined  PR† = ((ll22,,ll11,)...,lR,lR−1,−lR+1,lR+1) RR==21,4,... pmaetdhisataeclycowrdriitnegdtoowtnhteheabeoxvpeecptreedscnreiptt‘icohna,rgwee’accacnumimu--  (l2,l1,l4,l3,...,lR+1,lR) R=3,5,..(.3) ldaetfiendeonthtehefoluarttcicoem.pRoenfeenrrtisngoftothtehe2xke2rnmealtirnix(1a)s,wKecan σnσ1 Because P† is a unique permutation of P , the ensem- where the subscripts refer to the end and beginning di- R R ble, E , of all extended n-step paths P from the origin rections respectively. If the walker, starting in the +x F R is the same as the ensemble of all paths P† from the direction,loops overN entwined pairsand(x,t) is a lat- R origin. Furthermore, this is the same as the ensemble tice point within the light cone with t < tb then the of paths of the form (+1,σ2,σ3...) combined with all contribution to the +x-component of the net charge is orthogonal twins. Thus we may cover all paths in EF, proportional to ρ+ = N(K++(x,t)−K+−(x,t)). This with the correct Chessboard colouring, just by travers- is because an entwined loop corresponds to two forward ing all entwined pairs. This may be done through a sin- Chessboard paths, one originating from the origin with gle continuous (in the sense of the lattice) path since all a positive, blue first leg (K++ contribution) andthe sec- entwined loops intersect at the origin. Furthermore, en- ond from a negative red first leg(−K+− contribution). twined pairs fixed at the origin and at time t have the Similarly the −x-component at (x,t) is proportional to b same number of degrees of freedom as their individual ρ− = N(K−+(x,t)−K−−(x,t)). The ρ may be inter- component Chessboard paths (i.e. R−1) and each pair preted as particle densities which may be either positive may be given the statistical weight (ǫm)R−1 which cor- or negative depending on the predominance of entwined rectly weights the component Chessboard trajectories. trajectories in plus or minus t directions. ρ+ is positive Thus the following classical stochastic process gives rise in (+x,+t)-richareasandρ− is positive in (−x,+t)-rich to a properly weighted chessboard ensemble of coloured areas. Note that ρ+−ρ− is proportional to the sum of paths. Start a random walk at the origin and allow the the real and imaginary part of the Feynman propagator walker to choose entwined paths according to the num- ΦR+ ΦI(Gerschconventionforthe signofi). Unlikethe ber of free corners, either in the entwined path or one predecessors of this model[15, 16, 17], which did not use of the pairs. The walker traverses the entwined path as bound pairs of trajectories, this new model is relatively above so as to maintain both the Chessboard and time- easy to simulate on a lattice. sense colouring. The walker ends up at the origin at Fig.(3) shows an example of such a simulation, where the end of the traversal and repeats the process. The the sum of the real and imaginary parts of the propa- space-time lattice records the net number of traversals gator, Φ + Φ , at fixed t, are plotted versus x. The R I in the +t direction as the walker passes by registering a expected results from the Chessboard model are plotted plus one for a positive traversal and a minus one for a (continuous curve) at the same lattice resolution as the negative traversal, thus accumulating positive and neg- resultsofasimulationwithasinglepathwhichloopsover ative integers. The traversal weighting ensures that the the lattice 108 times. Inthe figure,t is 15steps fromthe constituent Chessboard paths have the correct expected origin,withanaverageoftwostepsbetweencornersora weight,and the ergodic nature ofthe walksinsures that, probabilityof1/2foradirectionchangeateachstep. At with enough loops, you can get as close as you like to a smaller values of t, the simulation and the Chessboard 4 model are indistinguishable on the scale of the figure, at tually(in a local ‘time’ parameter of the particle which largervaluesoftthesinglepathgivessparsercoverageof measures distance along the full space-time trajectory) the chessboard ensemble and the scatter increases. The forces the point particle which draws the propagator to individual real and imaginary parts of the propagator stay in the regiont>t , thus making it redraw the ‘fu- m may be calculated using the symmetry of the solutions, ture’ propagator in a manner that is consistent with an or by recording the ρ in two components to separate ‘initial’ condition at t = t . This change in the wave- ± m contributions from the original chessboard path and its function at t need not be unitary and may provide the m orthogonaltwin. analog of collapse. An interesting next step would be Although we do not know how much of the above can to see if a traversal and measurement scheme could be surviveinclusion ofanexternalfield and/orextensionto foundthatwouldinitiatecollapseinamannerconsistent three space dimensions, we do think the result reveals with the Born postulate. several qualitatively appealing features of the simplest case of a free particle in one dimension. First, the Feyn- man propagator has an independent existence as an ex- This work was partly funded by NSERC (GNO). The pected net charge over an ensemble of entwined paths authorsaregratefulforhelpfuldiscussionsandcomputa- which can be joined into a single trajectory. In this con- tional expertise from John Dorband and Scott Antonille text, the propagatorhas an underlying classicalstochas- at NASA-GSFC. tic model which is in effect self-quantizing and produces real densities in place of amplitudes. A second feature is that the above model provides a bridge between two distinct views of quantum mechan- ∗ corresponding author [email protected] ics in this case. Regarding Fig. 2, we may view the † [email protected] two trajectories in three ways. We can consider them [1] R. P. Feynman,Rev.Mod. Physics. 20, 367 (1948). astwoseparatechessboardtrajectories,colouredaccord- [2] R. P. Feynman and A. R. Hibbs, Quantum Mechanics ing to Feynman’s corner rule. An ensemble of such tra- and Path Integrals (New York: McGraw-Hill, 1965). jectories builds a quantum propagator as a sum-over- [3] S. Schweber, Reviewof Modern Physics 58, 449 (1986). histories. Thisistheconventionalview. Asecondpicture [4] S.Schweber,QEDandTheMenWhoMadeIt(Princeton is to note that an entwined pair forms a chain of cre- University Press, 1994). [5] H. Kroger, Phys.Rev.A. 55, 951 (1997). ation/annihilation events. An ensemble of these would [6] H. Kroger, Phys.Let. 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