ON THE HAMILTONIAN APPROACH AND PATH INTEGRATION FOR A POINT 4 PARTICLE MINIMALLY COUPLED TO 9 9 ELECTROMAGNETISM 1 n a Kostas Skenderis∗ and Peter van Nieuwenhuizen† J 6 Institute for Theoretical Physics, 1 State University of New York at Stony Brook, v 4 Stony Brook, NY 11794-3840 2 0 February 1, 2008 1 0 4 9 Abstract / h Wederivetheexactconfigurationspacepathintegral,togetherwith t - the way how to evaluate it, from the Hamiltonian approach for any p quantum mechanical system in flat spacetime whose Hamiltonian has e h at most two momentum operators. Starting from a given, covariant : or non-covariant, Hamiltonian, we go from the time-discretized path v i integralto the continuumpath integralby introducing Fourier modes. X We prove that the limit N for the terms in the perturbation → ∞ r expansion (“Feynman graphs”) exists, by demonstrating that the se- a ries involved are uniformly convergent. All terms in the expansion of the exponent in < x exp( ∆tHˆ/¯h)y > contribute to the propagator | − | (evenatorder∆t!). However,inthe time-discretizedpathintegralthe only effect of the terms with Hˆ2 and higher is to cancel terms which naivelyseemtovanishforN ,but,infact,arenonvanishing. The →∞ final result is that the naive correspondence between the Hamiltonian and the Lagrangian approach is correct, after all. We explicitly work through the example of a point particle coupled to electromagnetism. 2 Wecomputethepropagatortoorder(∆t) bothwiththeHamiltonian and the path integral approach and find agreement. ∗e-mail: [email protected] †e-mail: [email protected] 1 1 Introduction. Path integrals are often first written down in a symbolic way as an integral over paths of the exponent of an action, and then defined by some time- discretization. Of course, there are many ways in which to implement time- discretization. In some instances, rules have been discovered which lead to desirable answers for the path integral. A well-known example is the mid- point rule for the interaction dtA (x)x˙j of a point particle coupled to an j electromagnetic potential. This rule leads to gauge invariance of the terms R proportional to ∆t in the propagator[1] but it breaks gauge invariance in the terms of order (∆t)2. Theterms of higher order in ∆t are needed for the evaluation of anomalies (see below). In general, starting from a continuum path integral, there is no preferred way to discretize it. One might take the point of view that different discretizations simply correspond to different theories. In this article we take a different point of view. We take the Hamilto- nian formalism as starting point, and shall deduce both the action S config to be used in the configuration space path integral, and the way this path integral should be evaluated (“the measure”). We mean by the expressions “to be used” and “should be” that in this way the path integral formalism exactly reproducesthepropagator of theHamiltonian formalism. Of course, in the Hamiltonian Hˆ(xˆi,pˆ ) there is a priori a corresponding ambiguity in j the ordering of the operators xˆi and pˆ . However, in several examples, the j Hamiltonian of a quantum mechanical model is, in fact, the regulator of the Jacobians for symmetry transformations of a corresponding field theory, and these regulators are uniquely fixed by requiring certain symmetries of the field theory to be maintained at the quantum level[2, 3, 5, 6]. For ex- ample, in [3] regulators are constructed which maintain Weyl (local scale) invariance but as a consequence break Einstein (general coordinate) invari- ance. Thus: the field theory and the choice of which symmetries are free of anomalies fixes the regulator, the regulator is the Hamiltonian of a cor- responding quantum mechanical model, and the operator ordering of this Hamiltonian is thus fixed. For these reason we consider Hamiltonians of the formHˆ = pˆ2+ai(xˆ)pˆ +b(xˆ)whoseoperatororderingisfixedinthisway but i whose coefficents ai(xˆ) and b(xˆ) are not restricted except that we assume that they are regular functions; they may correspond to covariant or non- covariant Hamiltonians. The results of this paper prove which path integral (including, of course, the way how to evaluate it) corresponds to which reg- ulator (Hamiltonian). For chiral anomalies[2] this precise correspondence 2 was not needed due to their topological nature, but for trace anomalies[5, 6] and other anomalies of non-topological nature, the precise correspondence is crucial. Having obtained the 1-1 correspondence, it is then also possible to start with a particular action in the path integral (the latter to be evaluated as derived below) and to find the corresponding Hamiltonian operator. This will usually be the case when one is dealing with quantum field theories. For example, when one is dealing with renormalizable field theories or when the theory has certain symmetries the action may be known, and this will fix the operator ordering and the terms in the Hamiltonian. In the Hamiltonian approach the propagator is defined by ∆t < x,t y,t >=< x exp( Hˆ y >, ∆t = t t , (1) 2 1 2 1 | | − ¯h | − andevaluated, followingFeynman,byinsertingacompletesetofmomentum eigenstates ∆t <x,t y,t >= dp < x exp( Hˆ p >< p y > . (2) 2 1 | | − ¯h | | Z Expanding the exponent, and moving in each term ( ∆tHˆ/¯h)n/n! the xˆ − operators to the left and the pˆoperators to the right, one obtains an unam- biguous answer for the propagator. No regularization is needed. However, one must keep track of the commutators between xˆi and pˆ . It is often as- j sumed that it is sufficient to expand the exponent only to first order in ∆t, and to reexpontiate the result ∆t ∆t < x exp( Hˆ p > = exp[ h(x,p)] < xp > (false!) (3) | − ¯h | − ¯h | < xHˆ p > h(x,p) < xp > . (4) | | ≡ | This is incorrect for Hamiltonians with derivative coupling: for nonlinear sigma models where the pˆ2 term is multiplied by a function of xˆ (“the metric”)[7, 8, 9] or for Hamiltonians with a term A(xˆ) pˆ. We shall consider · the Hamiltonian 1 e e Hˆ = pˆ ( )A (xˆ) pˆi ( )Ai(xˆ) +V(xˆ), (5) i i 2 − c − c (cid:16) (cid:17)(cid:16) (cid:17) for arbitrary but nonsingular A (xˆ) and V(xˆ) which is obviously the most i general Hamiltonian of the form Hˆ = pˆ2 + ai(xˆ)pˆ + b(xˆ), and show that i 3 there are terms proportional to ∆t in the propagator which are due to com- mutators between pˆ and A (xˆ). In fact, all terms in the expansion of the i j exponential give such contributions[7, 8, 9]! Because the commutators [pˆi,xˆ ] = i¯hδi are proportional to h¯, the j − j propagator becomes a series in ¯h, ∆t and (x y)i with coefficients which are − functions of x. When we use the term “of order (∆t)k” we mean all terms which differ from the leading term by a factor (∆t)k, counting (x y)i as − (∆t)1/2. The terms of order h¯ w.r.t. the classical result correspond to one- loopcorrectionsinthepathintegralapproach,andcanbewrittenintermsof the classical action as the Van Vleck determinant[10, 11]. Terms of higher order in h¯ in the propagator can be computed straightforwardly (though tediously) in the Hamiltonian approach, again without need to specify a regularization. This indicates that the details of the path integral should follow straightforwardly from the Hamiltonian starting point. In particular it should not be necessary to fix a free constant in the overall normalization of the path integral by hand, for example by dividing by the path integral for a free particle. One begins by defining the path integral as N−1 N < x,0y, T >= lim dx < x ,t x ,t > , (6) α α−1 α−1 α α | − N→∞ | Z hαY=1 ihαY=1 i where x = x and x = y. This particular time-discretization follows from 0 N the Hamiltonian approach; it is due to the operator identity T T/N N exp( Hˆ)= exp( Hˆ) . (7) −¯h − ¯h (cid:16) (cid:17) The main result of this paper is a proof that the N limit exists, and → ∞ defines a continuum action S and an unambiguous and simple way to config evaluate the path integral perturbatively. We begin by splitting x into a background part z and a quantum part α α ξ . We shall also decompose the time-discretized action S into a part S α 0 which yields the propagator on the world line, and the rest which yields the interaction terms S . The z satisfy the equation of motion of S and int α 0 the boundary conditions z = y at α = N and z = x at α = 0, so that α α ξ = 0 both at α = 0 and at α = N. Since S is not equal to S, there α 0 are terms linear in ξ in the expansion of S(z +ξ,z˙ +ξ˙). Notice that the α time-discretized action S has not been obtained by some ad-hoc rule, but rather it is determined from the Hamiltonian approach. 4 The final result for the path integral should not depend on the choice of S . We choose S as the action of a free particle because that leads to 0 0 simple perturbation theory, but other choices of S should lead to the same 0 final result although the Feynman rules for the perturbative expansion of the path integral will be different. One now expands ξ into eigenfunctions of S , i.e., in terms of trigono- α 0 metric functions N−1 ξ = yksinαkπ/N, (α = 1,...,N 1). (8) α − k=1 X Changing integration variables from ξ to yk, the Jacobian is essentially α unity, while S is quadratic and diagonal in these yk. Rescaling these yk 0 such that the kinetic term in terms of the rescaled variables vk becomes the one of the continuum theory, the Jacobian of this rescaling leads to a non-trivial factor in the measure. At this point, the path integral has the generic form 1 dµexp[ (S +S (N))], (9) 0 int −¯h Z where the measure µ and the kinetic term S are already in the form of the 0 continuum theory, but the interaction S still depend on N. int By the term “continuum theory” we mean the path integral with (i) the classical Lagrangian L = 1x˙2 i(e)x˙iA +V, 2 − c i (ii) the expansion x(t) = z(t)+ ∞ vksinkπt/T, where z(t) is a solution k=1 of the equation of motion z¨= 0 with the boundary conditions z(0) = x and P z(T) = y, and (iii) the measure which normalizes the Gaussian integration with S over 0 the modes vk to (2π¯hT)−1/2 (not to unity because there is always one more intermediate set p >< p in Feynman’s approach than intermediate sets | | x >< x. The remaining factor (2π¯hT)−1/2 combines with classical part | | exp( (x y)2/2h¯T) to yield a representation of δ(x y) for small T). − − − One must then show that the limit N in S yields the interaction int → ∞ of the continuum theory. This is a well-known complicated problem, but we shall present here a totally elementary proof which uses only trigonometric relations such as 2sinαsinβ = cos(α β) cos(α+β) and the fact that − − the infinite series we encounter are uniformly convergent as functions of N. This property allows us to take the limit N before the summations → ∞ are performed. For the Hamiltonians of the form T(p)+V(x) such a proof is quite simple, but for non-vanishing vector potential A(x), we need rather laborious algebra. 5 The result is surprisingly simple. All terms in the propagator which in theHamiltonian approach aredueto commutators, areonly neededtomake sure that in the limit N one obtains the classical action. To be more → ∞ explicit, consider the discretized action in (40). The last three lines vanish in the naive limit N (the limit N for fixed mode index k) since → ∞ → ∞ they have extra factors 1/N w.r.t. the two lines above. These latter two lines naively yield the term dtA (x)x˙j in the classical action because 1/N j becomes dt and (ξ ξ ) becomes ξ˙dt. The claim is that if one does not α−1− α R take the naive limit but carefully evaluates the sums, then the non-naive terms in the first two lines cancel all of the last three lines in (40). To discuss in more detail what we mean by the naive limit N consider → ∞ the interaction terms N−1 N−1 1 S = vkvlλ(k)λ(l) sinαkπ/N sinαlπ/N, (10) int N k,l=1 α=1 X X where λ(k) = kπ[2N2(1 coskπ/N)]−1/2. For fixed k and N tending to − infinity one finds, λ(k) = 1 while for k N tending to infinity one has ∼ λ(k) = π/2. One may take the limit N in S (N) for given fixed naive → ∞ k and l, because the error thus committed cancels against the terms in S(N) S (N). Here S (N) is the time-discretized action we would naive naive − have obtained, if we had ignored the terms coming from commutators in the Hamiltonian approach. The non-trivial measure factorizes into a factor for each mode vk. One can theneasily computepropagators vkvl andFeynman graphsin termsof h i modes. Onecanalsousethequantum“fields”ξ(τ)andfindthat ξ(τ )ξ(τ ) 1 2 h i is theexpected worldlinepropagator (theinverse of∂2/∂τ2 withthecorrect boundary conditions). However, the mode representation is to be preffered because mode cut-off is the natural regularization scheme[5, 6]. Actually, all one-loop diagrams we evaluate are already finite by themselves since the divergences of the tadpole graphs are put to zero by mode regularization. Althoughwedonotconsiderherecurvedspace,wementionforcompleteness that in curved spacetime there are extra “ghosts” obtained by exponentiat- ing a factor (detg )1/2 in the measure and that with these ghosts all loop ij calculations become finite if one uses mode regularization[5, 6, 9]. From our point of view, the ambiguities often encountered in the def- inition of path integrals are due to starting “halfway”. Starting from the beginning, which means for us starting with the Hamiltonian approach, no ambiguities resultandone derivestheaction to beusedin thepath integral. 6 The result is the 1-1 correspondence 1 e e Hˆ = (pˆ ( )Aˆ ) δij (pˆ ( )Aˆ ) + Vˆ i i j j 2 − c − c (11) m 0 1 e S = dt[ δ x˙ix˙j i( )x˙iA + V], config ij i 2 − c Z−T and the path integral is perturbatively evaluated by computing Feynman graphs with given propagators and vertices. In section 2 we discuss the Hamiltonian approach for a point particle coupled to electromagnetism. Although we only need the propagator to order ∆t in order to construct the corresponding path integral, we evaluate ittoorder(∆t)2 inordertocomparelaterwithasimilarresultobtainedfrom thepathintegral. Ausefulcheck isthatitfactorizes intoaclassical partand a Van Vleck determinant. In section 3, the path integral is cast into a form wheretheonlyN-dependenceresidesintheinteractiontermsS . Insection int 4, we discuss the limit N in S . We organize the discussion by giving int → ∞ six examples which cover all possible cases one encounters in a perturbative evaluation of the path integral. In section 5 we evaluate as a check the path integral to order (∆t)2 at the one-loop level. Here we discuss how to evaluate the continuum path integrals in general in perturbation theory. The result agrees with the one obtained in section 2 from the Hamiltonian approach. In section 6 we note that our work straightforwardly extends to field theories with derivative coupling like Yang-Mills theory. We discuss how our work might be extended to curved spacetime, and also to phase space path integrals. 2 Hamiltonian operator approach. We wish to evaluate the propagator in Euclidean space < x exp( ∆tHˆ/¯h)y >, (12) | − | where 1 e e Hˆ = pˆ ( )A (xˆ) pˆi ( )Ai(xˆ) , i = 1,...,n. (13) i i 2 − c − c (cid:18) (cid:19)(cid:18) (cid:19) We donotaddatermV(x)sincetheanalysisfor thistermis thesameasfor the term (A (x))2. Indices are raised and lowered by δ , so for notational i ij 7 simplicity we write all indices down. We shall only use the commutation relation [pˆ,xˆ ] = ¯hδ and pˆ p >= p p > on momentum eigenstates p >. i j i ij i| i| | We rewrite the Hamiltonian as e e Hˆ = αˆ ( )βˆ+( )2γˆ, (14) − c c where 1 1 ¯hc αˆ = pˆ2, βˆ= Aˆ pˆ, γˆ = i( )∂ Aˆ+Aˆ2 . 2 · 2 e · (cid:18) (cid:19) Following Feynman we insert a complete set of p > states | < x exp( ∆tHˆ/¯h)y >= dnp < x exp( ∆tHˆ/¯h)p >< p y > . (15) | − | | − | | Z We expand the exponential and define 2k < x(Hˆ)k p >= Bk(x)pl < xp >, (16) | | l | l=0 X where Bk(x)pl is a polynomial of degree l in p’s, and l i < xp >= (2π¯h)−n/2exp( x p). (17) | ¯h · After rescaling the momenta as p = ¯h/∆tq we have p q (x y) <x exp( ∆tHˆ/¯h)y >= (4π2¯h∆t)−n/2 dnqexp(i · − ) | − | √¯h∆t Z ∞ ( 1)k 2k ∆t − Bk(x)ql( )k−l/2. (18) k! l ¯h k=0 l=0 X X The leading term comes from summing all the terms with l = 2k and has the simple form exp( 1q2). For this reason we introduced the q variable. −2 It follows that only a finite number of B’s need to be calculated in order to obtain the result up to desired order in ∆t. In particular, the result up to and including (∆t)2 needs the first five B’s (l = 2k through l = 2k 4). A − detailed discussion of the combinatorics is given in [9]. Here we merely give our result. Bk (x)q2k = αk, (19) 2k e Bk (x)q2k−1 = ( )kαk−1β, (20) 2k−1 − c 8 e Bk (x)q2k−2 = ( )2kαk−1γ + 2k−2 c e k i¯hc ( )2 αk−2 q (∂ β)+β2 , (21) c 2 ! "(cid:18) e (cid:19) i i # e k 1 i¯hc 2 i¯hc Bk (x)q2k−3 = ( )3 αk−2 ∂2β+ q ∂ γ 2k−3 − c 2 ! "2 (cid:18) e (cid:19) (cid:18) e (cid:19) i i i¯hc +2βγ + A (∂ A )q i i j j e # (cid:18) (cid:19) e k i¯hc 2 ( )3 αk−3 q q ∂ ∂ β − c 3 ! "(cid:18) e (cid:19) i j i j i¯hc +3 βq ∂ β+β3 , (22) i i (cid:18) e (cid:19) # e k 1 i¯hc 2 i¯hc Bk (x)q2k−4 = ( )4 αk−2 ∂2γ + A ∂ γ+γ2 2k−4 c 2 ! "2 (cid:18) e (cid:19) (cid:18) e (cid:19) i i # e k i¯hc 3 + ( )4 αk−3 q (∂ ∂2β) c 3 ! "(cid:18) e (cid:19) i i i¯hc 2 + [q q (∂ ∂ γ)+(∂ β)(∂ β) i j i j i i e (cid:18) (cid:19) 3 + β∂2β +q (∂ A )(∂ β)+2q A (∂ ∂ β)] i i j j i j i j 2 i¯hc + [3βq (∂ γ)+3γq (∂ β) i i i i e (cid:18) (cid:19) +3A (∂ β)β]+3β2γ i i # e k i¯hc 3 + ( )4 αk−4 q q q (∂ ∂ ∂ β) c 4 ! "(cid:18) e (cid:19) i j k i j k i¯hc 2 + [3(∂ β)(∂ β)q q +4(∂ ∂ β)βq q ] i j i j i j i j e (cid:18) (cid:19) i¯hc +6 β2(∂ β)q +β4 . (23) i i (cid:18) e (cid:19) # 9 The combinatorial factors k s ! indicate that only s out of k factors of Hˆ are involved in yielding commuta- tors. For example, the last term in (21) is due to picking two factors β and k 2 factors of α out of k factors Hˆ. Clearly this can be done in − k 2 ! ways and there are two powers of q less than in the leading term. Similarly, the one but last term in (21) is due to one commutator of αˆ and βˆ. Next we perform the summations over k which is easy and the Gaussian integrals which are straightforward but tedious. The result reads 1 < x exp( ∆tHˆ/¯h)y >= (2π¯h∆t)−n/2exp( (x y)2) | − | − 2∆t¯h − ie 1 − A (x y) i i ( −(cid:18) ¯hc (cid:19) − 1 ie ie 2 + − A + − A A (x y) (x y) i,j i j i j 2"(cid:18) ¯hc (cid:19) (cid:18) ¯hc (cid:19) # − − 1 ie ie 2 + − A +3 − A A i,jk i,j k 3!"(cid:18) ¯hc (cid:19) (cid:18) ¯hc (cid:19) ie 3 + − A A A (x y) (x y) (x y) i j k i j k (cid:18) ¯hc (cid:19) # − − − 1 ie ie 2 ie 2 + − A +3 − A A +4 − A A i,jkl i,j k,l i,jk l 4!"(cid:18) ¯hc (cid:19) (cid:18) ¯hc (cid:19) (cid:18) ¯hc (cid:19) ie 4 +6 − A A A A (x y) (x y) (x y) (x y) i j k l i j k l (cid:18) ¯hc (cid:19) # − − − − 1 ∆t e + ( )2F F (x y) (x y) ik kj i j 4! ¯h c − − i∆t e 1 + ( ) F (x y) F (x y) (x y) ki,k i ki,kj i j 12 c − − 2 − − h i 1 ∆t e (∆t)2 e ( )2A F (x y) (x y) ( )2F2 . (24) i kj,k i j −12 ¯h c − − − 48 c ) 10