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A note on Feynman's path integrals
Physica 60 (1972) 97-113
0 North-Holland Publishing Co.
A NOTE ON FEYNMAN’S M. J. GOOVAERTS* Rijksuniversitair
PATH INTEGRALS
and J. T. DEVREESE#
Centrum der Universiteit van Antwerpen, Antwerpen, Belgic
Received 31 August 1971
Synopsis Path integrals are introduced in this paper using the properties of integral equations of the Fredhohn type. This framework seems well adapted for studying the equivalence between the path-integral formalism and other formulations of quantum mechanics, and for examining the relation between path integrals and expansions like the Born series. The study of these integral equations shows how a density matrix is related to a Stieltjes integral. In principle our study reduces the problem of calculating a path integral to a problem of traditional analysis. A series expansion of a transform of the path integral which we introduced recently is shown to satisfy Schrodinger’s’equation and to be also related to a Liouville-type iteration of an integral equation. The problem of oscillatory integrals in Feyrmran’s formalism is also discussed.
1. Introduction. Feynman’s path-integral formulation of quantum mechanics becomes of increasing importance. First it is an appealing formulation from both the formal and intuitive point of view, because of its close relation to classical concepts and the impetus it provides for the study of functional analysis. Secondly its practical uses become more frequent and promising. The study of the iz transition in liquid He’) and the calculations on the polaron problem2*3p4) have been the first significant contributions. Practical applications have certainly been inhibited for a number of problems because path integrals had never been successful in treating fundamental problems like the hydrogen atom and the spin problem. Recently, however, progress has been made in both problemss*6*7). Also problems of e.g., statistical mechanics, are being treated now, be it numerically, by means of path integrals*). Large computers, properly used, certainly seem promising in solving numerically a number of quantum-mechanical problems; the advantage of the path-integral formulation being that one starts with a formal solution of the Schrodinger equation written as an infinite-tuple integral. A number of interesting problems arise if we consider the path-integral formalism: i) the problems concern* In partial fuliilment of the requirements for the Ph.D. ’ Also at the Solid State Physics Department, Studiecentrum
97
voor Kemenergie,
Mol, Belgi&.
98
M. J. GOOVAERTS AND J. T, DEVREESE
ing the exact mathematical meaning of path integrals; ii) the equivalence of Feynman’s formulation with historical formulations like Schrodinger’s formalism; iii) the problem of actually calculating path integrals (analytically or numerically). The problems of the exact mathematical meaning of. path ,integrals are related to normalization and the class of “trajectories” to be studied. They also involve the “convergence” problem arising from the well-known fact that oscillatory integrals occur in the path-integral formulation. Katog) has given a precise mathematical meaning to the Feynman path integral if the potential V in which the particle moves is sufficiently smooth. NelsonlO) has generalized the class of potentials considered by Kato. The study of the equivalence of path integrals and Schrodinger’s historical formulation does not seem to be settled in all generality. Blanc-Lapierre and Fortet’l) claim to have presented a quite sophisticated proof of this equivalence for a one-dimensional potential V(X). Recent progress has been made concerning the actual calculation of path integrals. Apart from the numerical calculations which are very promising we wish to bring up two recent analytical developments : Gutzwiller’s W.K.B. treatment of the Coulomb potentiaP) by means of path integrals (Garrod-Feynman type) which leads to the exact bound states. The promising feature of this treatment is that, conforming with some earlier studiesl’), only summations over classical paths have to be considered. It is true that the results of a W.K.B. method are not valid apriori, but path integrals have permitted the first W.K.B. treatment of the Coulomb potential. The expansion method introduced recently by the present authors which permitted the first straightforward analytic treatment of the hydrogen atom6) by means of path integrals and which at the same time has been applied to the polaron problem13$, the d-function potential14) and the harmonic potential15). This expansion method converts the Feynman path integral into a series of multiple integrals which is of the same type as a Liouville-Von Neuman solution for the Greenfunction problem. This expansion is a path integral written as a “traditional” mathematical expression. So we, comply with Gelfand’s and Yaglom’s statement, “Those results are very interesting which relate the evaluation of path integrals to more traditional mathematical problems”. The purpose of this paper is threefold. The “convergence” problem arising from oscillatory integrals is briefly discussed and it is shown ,how the oscillatory integrals can be avoided (section 2). The path-integral problem is developed from the properties of inhomogeneous integral equations (Fredholm type). Formally this derivation shows the equivalence of the path-integral formalism and Schrbdinger’s equation for a one-dimensional potential V(x). The formal resemblance with Born’s algebraic expansion will be stressed. However, we cannot claim absolute rigour because some interchanges of integrations and summations have not been proved and also because, as far as we
FEYNMAN’S
PATH INTEGRALS
99
know, problems might arise in defining the limit if n + 00 for the integral resulting after n iterations of the Liouville type (section 3). The properties of an expansion for a transform of the path integral, which is of the Liouville-Von Neuman type, will be examined. More specifically it will be shown that our integral transform satisfies the Schrijdinger equation (section 4). The present paper does not present new physical results. The considerations will neither obtain the rigour of highly mathematical workg) nor will there be a full discussion of the deep questions of the incorporation of a quantum-theoretical formalism into a complete mathematical theory16). Our purpose’is rather to discuss properties of path integrals from a new point of view (section 3), from computational considerations (section 4) or purely for “mise au point” (section 2). 2. On nonconvergent integrals in the path-integralformalism. In the path integral formulation of nonrelativistic quantum mechanics one encounters nonconvergent integrals, e.g., of the form +CO
s
x2 e’** dx. --m
(2.1)
Feynman defines these oscillatory integrals by using a convergence factor. These nonconvergent integrals arise, e.g., in Feynman’s argument concerning a single nonrelativistic particle with one degree of freedom17).
v(x2,td = +; k(xz,tz;xl,tl)~(xl,t,)dxl,
(2.2)
-CO
is a solution of Schriidinger’s equation if the kernel is defined as the following path integral k (x2, t2; x1, tl) = J eci’*)t! L(x’*‘t)dt Dx(t),
(2.3)
(where x = x1 if t = t, and x = x2 if t = tJ. In his discussion Feynman had to assume that
(2.4)
where m is the mass of the particle and A = (2xihejm)‘. He also assumed that the integrand only has to be considered up to second order in 7, because of the relation 7 w J& (which is based on intuitive grounds). As remarked also by Schweberl 8), “The lack of absolute convergence of (2.4) implies that mathematical
100
M. J. GOOVAERTS AND J.T. DEVREESE
difficulties are encountered in trying to give a rigorous meaning to the Feynman path integral in terms of a measure over a suitably defined function space (i.e. the space over all paths)“. We shall illustrate here how the difficulty of divergent integrals can be avoided after introducing “imaginary times”. In the case of a timeindependent one-dimensional potential V(x) the Schrbdinger equation with imaginary times takes the form:
--=aM
ap
-+s +
V(x)y
(2.5)
(where A = m = 1). The results can be generalized to the’case of a finite number of degrees of freedom and to any finite number of particles. First recall that the “Schriidinger equation” (2.5) has the same eigenvalues and eigenfunctions as H, the hamiltonian corresponding to (2.5) with real times. We now shall prove rigorously that every solution of the integral equation YJ(x, B + 4 =
j- x- (x, B + a; 6, B) P (6, B) d5, -CO
(2.6)
with I? (x, /I + E; 5, /?) defined as (’ 2,x”
- a,(+,
/3)],
(2.7)
satisfies Schriidinger’s equation. (E is an infinitesimal change of &) Making use of the equality +m
1
e-((X-0z/290) lim -C E-.O de (2x4* s
dx = ’ d2f(8 2 dt2
,
(2.8)
-a,
one finds after elementary manipulations
lim e+O
-!y(x, B + e) = hi $ ae
s
that the equation
15(x, B +
E;
5, ,Qy (E,B) dt ,
(2.9)
--m
written up to order E is equivalent with the “Schrodinger equation” (2.5). The argument given up to now only involves ,E(x, /3 + E; E, /I) for infinitesimal E. It is trivial to construct the path integral (for imaginary times) from k (x, p + E; E, j3). Applying (2.7) n times results in
FEYNMAN’S
101
PATH INTEGRALS
where k (xN, /? + NE; 5, ,Q)is defined as k(x,,/!?
+ Ne;5,/3) = ‘s” I?[+$ -cc
+ Ne;xN-l,B
... +POIt(x2,,3+2e;x~,/!? -co
+ (N - l)eldxN-1
+ e)It(x1,,9
+ e;5,/9)dxl.
Taking the limit for N -+ co and E + 0 with the restriction that NE = Aj3, a finite quantity, one obtains a path integral for imaginary times. In this section we have thus shown for a specific example that the difficulties related to the nonconvergent integrals appearing in the study of path integrals can be avoided by introducing the integration over the classical hamiltonian function E=jHdt=&?2dt+jV(x)dr, 0
0
0
and by calculating P = jemEDx(t), where in principle the same rules as in the case of usual path integrals, as introduced by Feynman have to be followed for the evaluation of P. 3. Study of the equivalence of Feynman's path-integralformalism and Schriidinger equation by means of inhomogeneous integral equations. As far as we know the equivalence of Feynman’s path-integral formalism and the “historical” treatments like the Schrijdinger formalism has never been proved rigorously (in all generality). Although FeynmanlQ), in his original paper, states that this is the case, his proof is based on defining nonconvergent integrals like (2.4). Schweber’*) has suggested the possibility that Feynman’s formulation is “somewhat more general than the historical one based on the correspondence between observables and linear selfadjoint operators and states to vectors in Hilbert space”. In this section we shall present a more rigorous proof of the equivalence of Feynman’s path-integral formulation and Schrodinger’s equation in the case of a particle moving in a one-dimensional potential V(x). Again extension to any finite number of particles and degrees of freedom is straightforward. Consider the following homogeneous integral equation (3.1) The kernel k: is defined by (2.7). This equation (3.1), if solved up to order E, corresponds to an eigenvalue problem which is equivalent to the Schriidinger equation
102
M. J. GOOVAERTS AND J.T. DEVREESE
for imaginary times. To study (3.1) we might convert this expression into a familiar inhomogeneous integral equation.
Y&) = do> + j F* 01,P’, 4) Y&‘) W.
(3.2)
This type of equation (equivalent to the Lippman-Schwinger equations) is well known from d-matrix theory; it expresses a wave function in the “representation” ,u in terms of representations ,u’. This expression is especially well adapted for constructing the algebraic Born series by successive iterations. These successive iterations lead to multiple integrals and it seems logical that path integrals might be related to these multiple integrals in the limit when the multiplicity tends to infinity. It is one of the purposes of this section to show that the “Born series” is indeed related to path integrals. Instead of transforming (3.1) into an inhomogeneous integral equation, we shall consider the artificial mathematical problem described by the following inhomogeneous integral equation e -(B+e)E
qx)
=
ei~x+
eeE
-02
+
E;
5, @)e-(B+e)E @p(t)dE.
(3.3)
This equation has eq. (3.1) as its corresponding homogeneous equation. The solution of this Fredholm-type integral equation20) by means of the LiouvilleVon Neuman method of successive substitutions and by a series will allow us to study the properties of k. It will be possible to express this kernel (also for finite A/?) as a path integral and to show the equivalence between the path-integral formalism and the Schrodinger equation for the system described by (3.1). First we solve (3.3) by series. Therefore et’“’ is developed as a convergent linear combination of the complete set of orthonormal “eigenfunctions” v.(x) of (3.1) which describes the physical system. (3.4) which equation de&es b,(p) as b,(p) = ‘f eigx Q)“(X)dx. -CO The solution of Fredholm’s equation (3.3) now takes the form (3.5) with 1 = eeE.
FEYNMAN’S
PATH ~INTEBRALS
103
Secondly we solve (3.3) using the Liouvihe-Von Neuman, method of successive substitutions giving +oo (3.6a) e -(B+e)E Q(X) = eiDX- 1 j k (x, 5, ;2)erpCd5, -00 where k (x, F, A) is the “reciprocal function” defined as k (x, E, A) = -K (x, B + E, 5, ,8) - mi2 I”-lk
(x, B + m.sE,P$,
(3.6b)
and k (x, j? + me; 5, @)is given by k(x,B
+ m&;&B) = +Jrnk[x,B
..* 7 1;(5z,B + z.,e:,B -CO
+ mc;L-l,B
+ E)E(t1,B
+ (m - 1)4&L-1
+~~;WG.
(3.7)
Note that the solution of (3.3) by the iteration procedure is formdy similar to an expansion of the Born type if an infinite number of terms is considered. However, the present treatment will avoid the well-known limitations of the Born approximation like Unite convergence radius of the expansion and occurrence of divergencies after a few iterations. It must also be emphasized .that the connection with the algebraic Born expansion is purely formal because the physics of the problem is contained in eq. (3.1) and not in eq. (3.3). Combining the solutions (3.5) and (3.6a) gives us the following relation: * b,(p) eeEn C cE R q&v) = eipx n=i e n-
iz
_r
k (x, 5, A) eip5d5.
(3.8)
If in (3.7) the limit m --, 00, E + 0 with rnc = A@(finite) is taken, the r.h.s. of this equation becomes a path integral (calculated for imaginary times) corresponding to a density matrix. We shah not deal here with the interesting but difficult problem of the “measure” which is adapted in that limit. In appendix A the convergence of path integrals associated with a class of potentials satisfying j 1V(r)1 d3r = finite, is discussed. It is realized that this class of potentials is very restricted and the appendix A may be considered as a first contribution to the study of wider classes of potentials. It is now possible to express k (x, @ + ms; E, /3) exclusively in terms of the eigenfunctions v”(x) and the energy eigenvalues En of eq. (3.1) which contain the physics of the problem being the eigenvalues and eigenfunctions of the Schrijdinger
104
M. J. GOOVAERTS AND J.T. DEVREESE
equation. For that purpose one multiplies both members of (3.6a) with il and performs h derivations with respect to the parameter 1. One is lead to
k (xvB + he; 5, B) =
-+$W,t,4l,=o.
(3.9)
From (3.8) one then finds for the reciprocal function
(3.10)
(3.4) then takes the form k(x,p
+ he;&/!?) =
+s
- V”(X)1
A=0
0
+m
x-
1
2x
s
b,(p) elpCdp .
(3.11)
After a few elementary steps k is then cast into the form k (x, B + ha; 5, B) = 5
eWheEn V.(X) v,(t).
(3.12)
II=1
Taking the limit for h + CQand E + 0 so that hs = Ap remains finite, one obtains k (x, B + AB; E, 8) = “i, e-ASEnV”(X)CM).
(3.13a)
Note that pin(x) is independent of h where k (x, /? + Ap; 5, /I) is now a path integral resulting from the performance of the limit. Also (3.13b) The r.h.s. of (3.13b) is known to be the density matrix or propagator for imaginary times of the system under study. It is equal to the path integral in the 1.h.s. Therefore the path integral contains all the physical information resulting from the Schrijdinger equation, while the Schriidinger equation permits the construction of the path integral. This shows the equivalence of the Schriidinger equation to the path-integral formalism for a particle moving in a one-dimensional potential I’(x).
FEYNMAN’S
PATH INTEGRALS
105
It now easily follows that: i) the kernel k (x, /?; 5, /I) satisfies Schriidinger’s equation for imaginary times inxandj3’; ii)limk(x,p + A/I’;&,t) = S(x - t); A/?+0
iii) the summation index 12can be a continuous variable as well as a discrete index. Eq. (3.5) has even the form of a Stieltjes transform. All the manipulations of interchanging summation and integral signs together with the derivations can be justified if one assumes sufficiently strong convergence for the density matrix which is examined in appendix A for a special class of potentials. 4. On an expansion for a transform of the path integral. In a study of the energy spectrum of the Coulomb-potential problem6) and of the polaron problem13) by means of path integrals we used an expansion for the integral transform W=
l
(4.la)
jd3r,k(rS,p;0,0),
and W=
i” dxsk(xsJ;O,O),
(4.lb)
--oo
of the path integral k for “imaginary times” /3. In the case of the d-function potentiali4) the method was extended by calculating the transform +m
s
ak(x,,B;xo,O),,=o. dxsxa ax
0
For the hydrogen atom the following expansion” is obtained W,
2 e2’
=
P($n
n=o . ..
B+n
-- 1
-I 1)
s
d3kn 1
(42)”
1 7tZns
- d3k, 1 + k:
1
1
1 + k,Z k: (k2 - k,)= ‘** (k, - k,,_l)= ’
* The more general expansion is W, = 2 eZn_$?-’ “=O d3k.
...
-s
(-
1
l)-”
d3kl 1 s (27~)~ kf exp [irO*(k, + ..* + k.)]
(2~)~ k; s(S+3~)...ts+3(kl+...+k.)=l’
where dp-’ denotes the inverse Laplace transform using the notations
(4.2)
106
M. J. GOOVAERTS
AND J. T. DEVREESE
This expansion has been obtained in a straightforward way starting from the path integral. This expansion has a close resemblance to the well known LiouvilleVon Neuman solution of the momentum-space integral equation for the Green function of the H atom2’) G(kz,k,,o)
=
63 (k, - k) k: - k2
2kv -
k: - k2
s
d3k3
4x
CW3 Ik, - k,l’
G(ks,k2,4, (4.3)
v = Ze2m/4xkh2.
However, the development (4.2) cannot be obtained directly from (4.3) by iteration. Indeed, G (k, , k, , o) is related to k (rp, /?; r,, a) via Fourier transformation over rs and r, , Once r, is set equal to zero and rg is integrated out G (k,,kl , co)can no longer be obtained from W,. The expansion of (4.la) has proved to be a useful procedure for actual calculations of the energy spectrum of exactly solvable systems. The generalization&r ref. 14 has also allowed one to obtain wave functions. Moreover, and this is the main purpose of the method, it seems reliable for the approximate study of quantum-mechanical systems. (This has been shown for the harmonic oscillator using the theory of moments and also for the polaron in the sense that expansions in the coupling constant and its inverse have been obtained.) Once an expansion like (4.2) has been obtained the path-integral part of the treatment is finished and classical analysis can be used from then on. We shall not deal here with the details of the derivation of the expansion (4.2) (section 4) from path integrals but wish to consider a generalization of this expansion which allows the study of wave functions and their properties. We shall distinguish between two different expansions which are generalizations of (4.la). One which is valid for quantum states with even wave functions P&J = lu(- rs)l and another which is valid for odd wave functions [(y(r& = -y( -r,.J]. In the former case we consider the expansion
W&o1 = +j=dqvW,d; -m
xc,,0).
(4.4)
In the latter case we consider the expansion
w, =
+; xsdx,k(x,,~;x,,O). -cQ
(4.5)
These expressions are written here again for a one-dimensional case which permits obvious generalizations. In ref. 14 these expressions have been used to study the d-function potential. In what follows it is our purpose to show that our expansions (4.4) and (4.5) are solutions of the Schrodinger equation for imaginary time. Of
FEYNMAN’S
course, this appear after integrations consistency
107
PATH INTEGRALS
has to be true a priori but if demonstrated on the expansions which numerous mathematical manipulations (interchanging of summations, and derivations) it shows the validity of our expansions and the of the treatment.
4.1. The case of even wave functions.
Our expansion for Ws[xo] is
with Q&9) = 9-l
+I) s -m +CO
. ..
s
exp [ix0 (k, + - *++ k.)]
%f(k.)
s (s+-)k:) a.. [s++ (k, -t ..- +k,)]*’
-CO
9-l is the usual notation for the inverse Laplace transform with respect to the variable s, resulting in a function of the parameter /?. The function f (k) denotes the Fourier transform of the potential V(x,-,)which is assumed to exist. Using the result of ref. 14 it is easily seen that
Qn(o> = 0,
n > 1.
Ws[xO] is written as
After some elementary calculations it is easily seen that +U3
l
a2Q, (19 = -Q._, 2 ax:
(/Cl)V(x,) + 9-l
s [S
$+k) -m
+Q,
...
s
$fM
exp [ix,, (k, -I- **- + k,)] s (s + +k:) --- [s + + (k, + . - - + kJ2
-co
Making use of the well-known formula for Laplace transforms 9-l
[&f(s)] = y,
if g-is(s)
= 9@)
and
V(O) = 0,
1 ’
108
M. J. GOOVAERTS AND J. T. DEVREESE
one obtains
1 a2en00 --= 2
axg
--%I@)
V(x,> + y.
Using this relation it is easily seen that Ws[x,,J satisfies the Schriidinger equation
a2~sL%l - V(x,) T ax; 1
4.2. The case comes
a Wxol. -&
W[x,]=
of odd
wave
+m
B
functions.
The expression for W,[k,]
I’,@)is defined as p&q
=
L1
ss
12! (27cB>f -53
xs dx,
dt,
0
. ..Sd~“~~...S~fl~~)...~(~“) 0
-co
--oo
- x0 + i i j=l
***+ k,ZT,,,, + 2k&,T,.,
k,t,12 + ix, jj kj - 3 WZ.1 J=l
+ *.. + 2kn_,3c,Tn_,,,]
,
and Ti,, is given by T i.J = 3 o* + 4) - -t I4 + cd* After a few elementary calculations I’,(/?) can be transformed into
+
be-
FEYNMAN’S
PATH INTEGRALS
109
One has
x
{s(s + &k: -I- iakl)
[s + 3 (k, + k$
+ ior (k, + k,]}-r
Working out the derivative with respect to OLone obtains as the final result for P&j?)
T”(B) being defined as 4-m T&l) = 9-l s --a,
2f(k,)
**. 1 %f(k.)
exp [ix,, (k, + **a+ k,)]
X s(s
Xi
f
+k:)+
[s + -)(k: + .a* + k,)2]
(k, + ... + k,)
~=1 [s + &(k, + ... + kj)2]’
It is easily proved that P&!?) satisfies
Using this relation one obtains
which again is the Schrijdinger equation for imaginary time. 5. Conclusions. It has been proved that the path integral-formalism can be introduced using the theory of Fredholm-type integral equations. The relation of this procedure with the Born series has been shown.
110
M. J. GOOVAERTS
AND
J.T. DEVREESE
The relation between a series expansion which we introduced recently6) for path integrals and a Liouville-type solution of a Fredholm equation has been studied, together with the properties of this series expansion. The convergence problem, arising from oscillatory integrals in Feynman’s path-integral formalism has been studied.
APPENDIX
A
As far as the convergence of
(A4
limk(x,B + me;t,B), m-m
is concerned, under the restriction that m be constant, one can remark that for the trivial case V(x) > 0 the following inequality holds: limk(x,j? m-rm
+ me;E,p) < limk,(x,@ m+a,
+ ms;E,B),
(A.3
where lim kf is the kernel of the free particle which is known. We shall now discuss m-ro.2 the convergence of a path integral associated with the class of potentials for which is finite.
j 1V(r)] d3r
(A-3)
This condition is sufficient for the existence of the Fourier transform of the potential. Expanding exp [ - jt V(r) dt] as a power series in the argument and writing [ - j”, V(r) dt] as an n-dimensional integral one obtains D(r,r,,,/?)
=~exp(tSizdt)Dr(l)~~~~~dll 0
0 B
.*a
ss dt,
0
d3kif(kl) ... w3
s
sf(kJ
exp(ijiI
5r ($)),
(A.4)
where D (up, r. , p) is a notation for the path integral D(ra,ro,p)
= Jexp[-+o~ildf
- iV(r)dt]Dr(t),
(-4.5)
f(k) and V(k) are related by d3kf(k) eikSr, V(r) = 1 (W3 s
(A*6)
FEYNMAN’S
PATH INTEGRALS
111
and f(k) = j V(r) eek*’ d3r.
(A-7)
Taking into account convergence arguments it can be proved that interchanging the path-integral sign and the other operator signs is a legitimate manipulation. Define I. as I,, = j exp 1-3 i P dt f !.&(t) r(t) dll Dr (t),
64.8)
fn(f) = l;iI kj 6 (2 - t,).
(A-9)
with
This integral is a gaussian one and can be calculated using the method suggested by Feynman, resulting in 1 -- 1 (q - r. - i i Ic,~~)~+ irO i kJ -j=L j=l 2/3 In - (27$3)+ exp I
(A.lO) with Ti,, defined as
We now aim at calculating
W(ro,B>= jd3r~~(r~,ro,L06).
(A.13
The manipulations to be performed are elementary because of the uniform convergence of the series development for D (r@, ro, /?) holding for every rp. One obtains B B
x exp [-+(k:T,,,
+ *** + 2kn-lknTn-~,n)3.
(A.13)
112
M. J. GOOVAERTS AND J. T. DEVREESE
Calculating obtained
the Laplace transform
...
s
of both members the following result is
J2)
d3knf(kn
explib ’ ckl +
‘*’
+
‘“)I.
s (s + k:) -*a (s + k,Z)
(A.14)
For Re (s) > 0 one obtains the following inequality d3k, If(kl J2)] 1”.
(A.15)
s
Taking into account only s values satisfying Re (s) > 0 and
IsI > j d3k, If(b &)I it
(A.16)
3
follows that is finite.
The same statement respect to s
(A.17)
can easily be proved for the derivations
is finite
i=
of (A.14) with
1,2,....
So as soon as Re (s) is sufficiently large Y [ W(r,, /I), s] is an analytic function of s and W (r. , #I) has to be finite except for an infinite or finite number of discrete J!?values.
REFERENCES 1) Feynman, R.P., Phys. Rev. 91 (1953) 1291. 2) Feynman, R. P., Phys. Rev. 97 (1955) 669. 3) Feynman, R.P., Hellwarth, R. W., Iddings, C. K. and Platzman, P.M., Phys. Rev. 127 (1962) 1004. 4) Platzman, P.M., Phys. Rev. 125 (1962) 1961. 5) Gutzwiller, M., J. math. Phys. 8 (1967) 1979. 6) Goovaerts, M. and Devreese, J., J. math. Phys., to be published. 7) Schulman, L., Phys. Rev. 176 (1968) 1558.
FEYNMAN’S 8) 9) 10) 11)
PATH INTEGRALS
113
Fosdick, L.D. and Jordan, H. J., Phys. Rev. 143 (1966) 58. Kato, T., Trans. Amer. Math. Sot. 70 (1951) 195. Nelson, E., J. math. Phys. 5 (1964) 332. Blanc-Lapierre, A. and Fortet, R., Theorie des Fonctions aleatbires, Mason et Cie (Paris, 1953). 12) Clutton-Brock, M., Proc. Cambridge Phil. Sot. 61(1965) 201. 13) Devreese, J. and Goovaerts, M., A Modified Perturbation Development for Energy States. Application to the Calculation of Self-energy, and Effective Mass of an Electron in a Polar Crystal, Internal report, to be published. 14) Goovaerts, M., Babcenco, A. and Devreese, J., A New Expansion Method in the Feynman Path-integral Formalism. Application to a One-dimensional Delta-function Potential, Internal Report, to be published. 15) Goovaerts, M., Thesis, Ghent, 1971. 16) Antoine, J.-P., J. math. Phys. (1960) 53, (1969) 2276. 17) Feynman, R.P. and Hibbs, A.R., Quantum Mechanics and Path-Integrals, McGraw-Hill Book Co (New York, 1965), p. 76. 18) Schweber, S.S., J. math. Phys. 3 (1962) 831. 19) Feynman, R.P., Rev. mod. Phys. 20 (1948) 367. 20) Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis (eds.) Cambridge at the University Press (London, 1965), p.213. 21) Schwinger, J., J. math. Phys. 5 (1964) 1606.
0 North-Holland Publishing Co.
A NOTE ON FEYNMAN’S M. J. GOOVAERTS* Rijksuniversitair
PATH INTEGRALS
and J. T. DEVREESE#
Centrum der Universiteit van Antwerpen, Antwerpen, Belgic
Received 31 August 1971
Synopsis Path integrals are introduced in this paper using the properties of integral equations of the Fredhohn type. This framework seems well adapted for studying the equivalence between the path-integral formalism and other formulations of quantum mechanics, and for examining the relation between path integrals and expansions like the Born series. The study of these integral equations shows how a density matrix is related to a Stieltjes integral. In principle our study reduces the problem of calculating a path integral to a problem of traditional analysis. A series expansion of a transform of the path integral which we introduced recently is shown to satisfy Schrodinger’s’equation and to be also related to a Liouville-type iteration of an integral equation. The problem of oscillatory integrals in Feyrmran’s formalism is also discussed.
1. Introduction. Feynman’s path-integral formulation of quantum mechanics becomes of increasing importance. First it is an appealing formulation from both the formal and intuitive point of view, because of its close relation to classical concepts and the impetus it provides for the study of functional analysis. Secondly its practical uses become more frequent and promising. The study of the iz transition in liquid He’) and the calculations on the polaron problem2*3p4) have been the first significant contributions. Practical applications have certainly been inhibited for a number of problems because path integrals had never been successful in treating fundamental problems like the hydrogen atom and the spin problem. Recently, however, progress has been made in both problemss*6*7). Also problems of e.g., statistical mechanics, are being treated now, be it numerically, by means of path integrals*). Large computers, properly used, certainly seem promising in solving numerically a number of quantum-mechanical problems; the advantage of the path-integral formulation being that one starts with a formal solution of the Schrodinger equation written as an infinite-tuple integral. A number of interesting problems arise if we consider the path-integral formalism: i) the problems concern* In partial fuliilment of the requirements for the Ph.D. ’ Also at the Solid State Physics Department, Studiecentrum
97
voor Kemenergie,
Mol, Belgi&.
98
M. J. GOOVAERTS AND J. T, DEVREESE
ing the exact mathematical meaning of path integrals; ii) the equivalence of Feynman’s formulation with historical formulations like Schrodinger’s formalism; iii) the problem of actually calculating path integrals (analytically or numerically). The problems of the exact mathematical meaning of. path ,integrals are related to normalization and the class of “trajectories” to be studied. They also involve the “convergence” problem arising from the well-known fact that oscillatory integrals occur in the path-integral formulation. Katog) has given a precise mathematical meaning to the Feynman path integral if the potential V in which the particle moves is sufficiently smooth. NelsonlO) has generalized the class of potentials considered by Kato. The study of the equivalence of path integrals and Schrodinger’s historical formulation does not seem to be settled in all generality. Blanc-Lapierre and Fortet’l) claim to have presented a quite sophisticated proof of this equivalence for a one-dimensional potential V(X). Recent progress has been made concerning the actual calculation of path integrals. Apart from the numerical calculations which are very promising we wish to bring up two recent analytical developments : Gutzwiller’s W.K.B. treatment of the Coulomb potentiaP) by means of path integrals (Garrod-Feynman type) which leads to the exact bound states. The promising feature of this treatment is that, conforming with some earlier studiesl’), only summations over classical paths have to be considered. It is true that the results of a W.K.B. method are not valid apriori, but path integrals have permitted the first W.K.B. treatment of the Coulomb potential. The expansion method introduced recently by the present authors which permitted the first straightforward analytic treatment of the hydrogen atom6) by means of path integrals and which at the same time has been applied to the polaron problem13$, the d-function potential14) and the harmonic potential15). This expansion method converts the Feynman path integral into a series of multiple integrals which is of the same type as a Liouville-Von Neuman solution for the Greenfunction problem. This expansion is a path integral written as a “traditional” mathematical expression. So we, comply with Gelfand’s and Yaglom’s statement, “Those results are very interesting which relate the evaluation of path integrals to more traditional mathematical problems”. The purpose of this paper is threefold. The “convergence” problem arising from oscillatory integrals is briefly discussed and it is shown ,how the oscillatory integrals can be avoided (section 2). The path-integral problem is developed from the properties of inhomogeneous integral equations (Fredholm type). Formally this derivation shows the equivalence of the path-integral formalism and Schrbdinger’s equation for a one-dimensional potential V(x). The formal resemblance with Born’s algebraic expansion will be stressed. However, we cannot claim absolute rigour because some interchanges of integrations and summations have not been proved and also because, as far as we
FEYNMAN’S
PATH INTEGRALS
99
know, problems might arise in defining the limit if n + 00 for the integral resulting after n iterations of the Liouville type (section 3). The properties of an expansion for a transform of the path integral, which is of the Liouville-Von Neuman type, will be examined. More specifically it will be shown that our integral transform satisfies the Schrijdinger equation (section 4). The present paper does not present new physical results. The considerations will neither obtain the rigour of highly mathematical workg) nor will there be a full discussion of the deep questions of the incorporation of a quantum-theoretical formalism into a complete mathematical theory16). Our purpose’is rather to discuss properties of path integrals from a new point of view (section 3), from computational considerations (section 4) or purely for “mise au point” (section 2). 2. On nonconvergent integrals in the path-integralformalism. In the path integral formulation of nonrelativistic quantum mechanics one encounters nonconvergent integrals, e.g., of the form +CO
s
x2 e’** dx. --m
(2.1)
Feynman defines these oscillatory integrals by using a convergence factor. These nonconvergent integrals arise, e.g., in Feynman’s argument concerning a single nonrelativistic particle with one degree of freedom17).
v(x2,td = +; k(xz,tz;xl,tl)~(xl,t,)dxl,
(2.2)
-CO
is a solution of Schriidinger’s equation if the kernel is defined as the following path integral k (x2, t2; x1, tl) = J eci’*)t! L(x’*‘t)dt Dx(t),
(2.3)
(where x = x1 if t = t, and x = x2 if t = tJ. In his discussion Feynman had to assume that
(2.4)
where m is the mass of the particle and A = (2xihejm)‘. He also assumed that the integrand only has to be considered up to second order in 7, because of the relation 7 w J& (which is based on intuitive grounds). As remarked also by Schweberl 8), “The lack of absolute convergence of (2.4) implies that mathematical
100
M. J. GOOVAERTS AND J.T. DEVREESE
difficulties are encountered in trying to give a rigorous meaning to the Feynman path integral in terms of a measure over a suitably defined function space (i.e. the space over all paths)“. We shall illustrate here how the difficulty of divergent integrals can be avoided after introducing “imaginary times”. In the case of a timeindependent one-dimensional potential V(x) the Schrbdinger equation with imaginary times takes the form:
--=aM
ap
-+s +
V(x)y
(2.5)
(where A = m = 1). The results can be generalized to the’case of a finite number of degrees of freedom and to any finite number of particles. First recall that the “Schriidinger equation” (2.5) has the same eigenvalues and eigenfunctions as H, the hamiltonian corresponding to (2.5) with real times. We now shall prove rigorously that every solution of the integral equation YJ(x, B + 4 =
j- x- (x, B + a; 6, B) P (6, B) d5, -CO
(2.6)
with I? (x, /I + E; 5, /?) defined as (’ 2,x”
- a,(+,
/3)],
(2.7)
satisfies Schriidinger’s equation. (E is an infinitesimal change of &) Making use of the equality +m
1
e-((X-0z/290) lim -C E-.O de (2x4* s
dx = ’ d2f(8 2 dt2
,
(2.8)
-a,
one finds after elementary manipulations
lim e+O
-!y(x, B + e) = hi $ ae
s
that the equation
15(x, B +
E;
5, ,Qy (E,B) dt ,
(2.9)
--m
written up to order E is equivalent with the “Schrodinger equation” (2.5). The argument given up to now only involves ,E(x, /3 + E; E, /I) for infinitesimal E. It is trivial to construct the path integral (for imaginary times) from k (x, p + E; E, j3). Applying (2.7) n times results in
FEYNMAN’S
101
PATH INTEGRALS
where k (xN, /? + NE; 5, ,Q)is defined as k(x,,/!?
+ Ne;5,/3) = ‘s” I?[+$ -cc
+ Ne;xN-l,B
... +POIt(x2,,3+2e;x~,/!? -co
+ (N - l)eldxN-1
+ e)It(x1,,9
+ e;5,/9)dxl.
Taking the limit for N -+ co and E + 0 with the restriction that NE = Aj3, a finite quantity, one obtains a path integral for imaginary times. In this section we have thus shown for a specific example that the difficulties related to the nonconvergent integrals appearing in the study of path integrals can be avoided by introducing the integration over the classical hamiltonian function E=jHdt=&?2dt+jV(x)dr, 0
0
0
and by calculating P = jemEDx(t), where in principle the same rules as in the case of usual path integrals, as introduced by Feynman have to be followed for the evaluation of P. 3. Study of the equivalence of Feynman's path-integralformalism and Schriidinger equation by means of inhomogeneous integral equations. As far as we know the equivalence of Feynman’s path-integral formalism and the “historical” treatments like the Schrijdinger formalism has never been proved rigorously (in all generality). Although FeynmanlQ), in his original paper, states that this is the case, his proof is based on defining nonconvergent integrals like (2.4). Schweber’*) has suggested the possibility that Feynman’s formulation is “somewhat more general than the historical one based on the correspondence between observables and linear selfadjoint operators and states to vectors in Hilbert space”. In this section we shall present a more rigorous proof of the equivalence of Feynman’s path-integral formulation and Schrodinger’s equation in the case of a particle moving in a one-dimensional potential V(x). Again extension to any finite number of particles and degrees of freedom is straightforward. Consider the following homogeneous integral equation (3.1) The kernel k: is defined by (2.7). This equation (3.1), if solved up to order E, corresponds to an eigenvalue problem which is equivalent to the Schriidinger equation
102
M. J. GOOVAERTS AND J.T. DEVREESE
for imaginary times. To study (3.1) we might convert this expression into a familiar inhomogeneous integral equation.
Y&) = do> + j F* 01,P’, 4) Y&‘) W.
(3.2)
This type of equation (equivalent to the Lippman-Schwinger equations) is well known from d-matrix theory; it expresses a wave function in the “representation” ,u in terms of representations ,u’. This expression is especially well adapted for constructing the algebraic Born series by successive iterations. These successive iterations lead to multiple integrals and it seems logical that path integrals might be related to these multiple integrals in the limit when the multiplicity tends to infinity. It is one of the purposes of this section to show that the “Born series” is indeed related to path integrals. Instead of transforming (3.1) into an inhomogeneous integral equation, we shall consider the artificial mathematical problem described by the following inhomogeneous integral equation e -(B+e)E
qx)
=
ei~x+
eeE
-02
+
E;
5, @)e-(B+e)E @p(t)dE.
(3.3)
This equation has eq. (3.1) as its corresponding homogeneous equation. The solution of this Fredholm-type integral equation20) by means of the LiouvilleVon Neuman method of successive substitutions and by a series will allow us to study the properties of k. It will be possible to express this kernel (also for finite A/?) as a path integral and to show the equivalence between the path-integral formalism and the Schrodinger equation for the system described by (3.1). First we solve (3.3) by series. Therefore et’“’ is developed as a convergent linear combination of the complete set of orthonormal “eigenfunctions” v.(x) of (3.1) which describes the physical system. (3.4) which equation de&es b,(p) as b,(p) = ‘f eigx Q)“(X)dx. -CO The solution of Fredholm’s equation (3.3) now takes the form (3.5) with 1 = eeE.
FEYNMAN’S
PATH ~INTEBRALS
103
Secondly we solve (3.3) using the Liouvihe-Von Neuman, method of successive substitutions giving +oo (3.6a) e -(B+e)E Q(X) = eiDX- 1 j k (x, 5, ;2)erpCd5, -00 where k (x, F, A) is the “reciprocal function” defined as k (x, E, A) = -K (x, B + E, 5, ,8) - mi2 I”-lk
(x, B + m.sE,P$,
(3.6b)
and k (x, j? + me; 5, @)is given by k(x,B
+ m&;&B) = +Jrnk[x,B
..* 7 1;(5z,B + z.,e:,B -CO
+ mc;L-l,B
+ E)E(t1,B
+ (m - 1)4&L-1
+~~;WG.
(3.7)
Note that the solution of (3.3) by the iteration procedure is formdy similar to an expansion of the Born type if an infinite number of terms is considered. However, the present treatment will avoid the well-known limitations of the Born approximation like Unite convergence radius of the expansion and occurrence of divergencies after a few iterations. It must also be emphasized .that the connection with the algebraic Born expansion is purely formal because the physics of the problem is contained in eq. (3.1) and not in eq. (3.3). Combining the solutions (3.5) and (3.6a) gives us the following relation: * b,(p) eeEn C cE R q&v) = eipx n=i e n-
iz
_r
k (x, 5, A) eip5d5.
(3.8)
If in (3.7) the limit m --, 00, E + 0 with rnc = A@(finite) is taken, the r.h.s. of this equation becomes a path integral (calculated for imaginary times) corresponding to a density matrix. We shah not deal here with the interesting but difficult problem of the “measure” which is adapted in that limit. In appendix A the convergence of path integrals associated with a class of potentials satisfying j 1V(r)1 d3r = finite, is discussed. It is realized that this class of potentials is very restricted and the appendix A may be considered as a first contribution to the study of wider classes of potentials. It is now possible to express k (x, @ + ms; E, /3) exclusively in terms of the eigenfunctions v”(x) and the energy eigenvalues En of eq. (3.1) which contain the physics of the problem being the eigenvalues and eigenfunctions of the Schrijdinger
104
M. J. GOOVAERTS AND J.T. DEVREESE
equation. For that purpose one multiplies both members of (3.6a) with il and performs h derivations with respect to the parameter 1. One is lead to
k (xvB + he; 5, B) =
-+$W,t,4l,=o.
(3.9)
From (3.8) one then finds for the reciprocal function
(3.10)
(3.4) then takes the form k(x,p
+ he;&/!?) =
+s
- V”(X)1
A=0
0
+m
x-
1
2x
s
b,(p) elpCdp .
(3.11)
After a few elementary steps k is then cast into the form k (x, B + ha; 5, B) = 5
eWheEn V.(X) v,(t).
(3.12)
II=1
Taking the limit for h + CQand E + 0 so that hs = Ap remains finite, one obtains k (x, B + AB; E, 8) = “i, e-ASEnV”(X)CM).
(3.13a)
Note that pin(x) is independent of h where k (x, /? + Ap; 5, /I) is now a path integral resulting from the performance of the limit. Also (3.13b) The r.h.s. of (3.13b) is known to be the density matrix or propagator for imaginary times of the system under study. It is equal to the path integral in the 1.h.s. Therefore the path integral contains all the physical information resulting from the Schrijdinger equation, while the Schriidinger equation permits the construction of the path integral. This shows the equivalence of the Schriidinger equation to the path-integral formalism for a particle moving in a one-dimensional potential I’(x).
FEYNMAN’S
PATH INTEGRALS
105
It now easily follows that: i) the kernel k (x, /?; 5, /I) satisfies Schriidinger’s equation for imaginary times inxandj3’; ii)limk(x,p + A/I’;&,t) = S(x - t); A/?+0
iii) the summation index 12can be a continuous variable as well as a discrete index. Eq. (3.5) has even the form of a Stieltjes transform. All the manipulations of interchanging summation and integral signs together with the derivations can be justified if one assumes sufficiently strong convergence for the density matrix which is examined in appendix A for a special class of potentials. 4. On an expansion for a transform of the path integral. In a study of the energy spectrum of the Coulomb-potential problem6) and of the polaron problem13) by means of path integrals we used an expansion for the integral transform W=
l
(4.la)
jd3r,k(rS,p;0,0),
and W=
i” dxsk(xsJ;O,O),
(4.lb)
--oo
of the path integral k for “imaginary times” /3. In the case of the d-function potentiali4) the method was extended by calculating the transform +m
s
ak(x,,B;xo,O),,=o. dxsxa ax
0
For the hydrogen atom the following expansion” is obtained W,
2 e2’
=
P($n
n=o . ..
B+n
-- 1
-I 1)
s
d3kn 1
(42)”
1 7tZns
- d3k, 1 + k:
1
1
1 + k,Z k: (k2 - k,)= ‘** (k, - k,,_l)= ’
* The more general expansion is W, = 2 eZn_$?-’ “=O d3k.
...
-s
(-
1
l)-”
d3kl 1 s (27~)~ kf exp [irO*(k, + ..* + k.)]
(2~)~ k; s(S+3~)...ts+3(kl+...+k.)=l’
where dp-’ denotes the inverse Laplace transform using the notations
(4.2)
106
M. J. GOOVAERTS
AND J. T. DEVREESE
This expansion has been obtained in a straightforward way starting from the path integral. This expansion has a close resemblance to the well known LiouvilleVon Neuman solution of the momentum-space integral equation for the Green function of the H atom2’) G(kz,k,,o)
=
63 (k, - k) k: - k2
2kv -
k: - k2
s
d3k3
4x
CW3 Ik, - k,l’
G(ks,k2,4, (4.3)
v = Ze2m/4xkh2.
However, the development (4.2) cannot be obtained directly from (4.3) by iteration. Indeed, G (k, , k, , o) is related to k (rp, /?; r,, a) via Fourier transformation over rs and r, , Once r, is set equal to zero and rg is integrated out G (k,,kl , co)can no longer be obtained from W,. The expansion of (4.la) has proved to be a useful procedure for actual calculations of the energy spectrum of exactly solvable systems. The generalization&r ref. 14 has also allowed one to obtain wave functions. Moreover, and this is the main purpose of the method, it seems reliable for the approximate study of quantum-mechanical systems. (This has been shown for the harmonic oscillator using the theory of moments and also for the polaron in the sense that expansions in the coupling constant and its inverse have been obtained.) Once an expansion like (4.2) has been obtained the path-integral part of the treatment is finished and classical analysis can be used from then on. We shall not deal here with the details of the derivation of the expansion (4.2) (section 4) from path integrals but wish to consider a generalization of this expansion which allows the study of wave functions and their properties. We shall distinguish between two different expansions which are generalizations of (4.la). One which is valid for quantum states with even wave functions P&J = lu(- rs)l and another which is valid for odd wave functions [(y(r& = -y( -r,.J]. In the former case we consider the expansion
W&o1 = +j=dqvW,d; -m
xc,,0).
(4.4)
In the latter case we consider the expansion
w, =
+; xsdx,k(x,,~;x,,O). -cQ
(4.5)
These expressions are written here again for a one-dimensional case which permits obvious generalizations. In ref. 14 these expressions have been used to study the d-function potential. In what follows it is our purpose to show that our expansions (4.4) and (4.5) are solutions of the Schrodinger equation for imaginary time. Of
FEYNMAN’S
course, this appear after integrations consistency
107
PATH INTEGRALS
has to be true a priori but if demonstrated on the expansions which numerous mathematical manipulations (interchanging of summations, and derivations) it shows the validity of our expansions and the of the treatment.
4.1. The case of even wave functions.
Our expansion for Ws[xo] is
with Q&9) = 9-l
+I) s -m +CO
. ..
s
exp [ix0 (k, + - *++ k.)]
%f(k.)
s (s+-)k:) a.. [s++ (k, -t ..- +k,)]*’
-CO
9-l is the usual notation for the inverse Laplace transform with respect to the variable s, resulting in a function of the parameter /?. The function f (k) denotes the Fourier transform of the potential V(x,-,)which is assumed to exist. Using the result of ref. 14 it is easily seen that
Qn(o> = 0,
n > 1.
Ws[xO] is written as
After some elementary calculations it is easily seen that +U3
l
a2Q, (19 = -Q._, 2 ax:
(/Cl)V(x,) + 9-l
s [S
$+k) -m
+Q,
...
s
$fM
exp [ix,, (k, -I- **- + k,)] s (s + +k:) --- [s + + (k, + . - - + kJ2
-co
Making use of the well-known formula for Laplace transforms 9-l
[&f(s)] = y,
if g-is(s)
= 9@)
and
V(O) = 0,
1 ’
108
M. J. GOOVAERTS AND J. T. DEVREESE
one obtains
1 a2en00 --= 2
axg
--%I@)
V(x,> + y.
Using this relation it is easily seen that Ws[x,,J satisfies the Schriidinger equation
a2~sL%l - V(x,) T ax; 1
4.2. The case comes
a Wxol. -&
W[x,]=
of odd
wave
+m
B
functions.
The expression for W,[k,]
I’,@)is defined as p&q
=
L1
ss
12! (27cB>f -53
xs dx,
dt,
0
. ..Sd~“~~...S~fl~~)...~(~“) 0
-co
--oo
- x0 + i i j=l
***+ k,ZT,,,, + 2k&,T,.,
k,t,12 + ix, jj kj - 3 WZ.1 J=l
+ *.. + 2kn_,3c,Tn_,,,]
,
and Ti,, is given by T i.J = 3 o* + 4) - -t I4 + cd* After a few elementary calculations I’,(/?) can be transformed into
+
be-
FEYNMAN’S
PATH INTEGRALS
109
One has
x
{s(s + &k: -I- iakl)
[s + 3 (k, + k$
+ ior (k, + k,]}-r
Working out the derivative with respect to OLone obtains as the final result for P&j?)
T”(B) being defined as 4-m T&l) = 9-l s --a,
2f(k,)
**. 1 %f(k.)
exp [ix,, (k, + **a+ k,)]
X s(s
Xi
f
+k:)+
[s + -)(k: + .a* + k,)2]
(k, + ... + k,)
~=1 [s + &(k, + ... + kj)2]’
It is easily proved that P&!?) satisfies
Using this relation one obtains
which again is the Schrijdinger equation for imaginary time. 5. Conclusions. It has been proved that the path integral-formalism can be introduced using the theory of Fredholm-type integral equations. The relation of this procedure with the Born series has been shown.
110
M. J. GOOVAERTS
AND
J.T. DEVREESE
The relation between a series expansion which we introduced recently6) for path integrals and a Liouville-type solution of a Fredholm equation has been studied, together with the properties of this series expansion. The convergence problem, arising from oscillatory integrals in Feynman’s path-integral formalism has been studied.
APPENDIX
A
As far as the convergence of
(A4
limk(x,B + me;t,B), m-m
is concerned, under the restriction that m be constant, one can remark that for the trivial case V(x) > 0 the following inequality holds: limk(x,j? m-rm
+ me;E,p) < limk,(x,@ m+a,
+ ms;E,B),
(A.3
where lim kf is the kernel of the free particle which is known. We shall now discuss m-ro.2 the convergence of a path integral associated with the class of potentials for which is finite.
j 1V(r)] d3r
(A-3)
This condition is sufficient for the existence of the Fourier transform of the potential. Expanding exp [ - jt V(r) dt] as a power series in the argument and writing [ - j”, V(r) dt] as an n-dimensional integral one obtains D(r,r,,,/?)
=~exp(tSizdt)Dr(l)~~~~~dll 0
0 B
.*a
ss dt,
0
d3kif(kl) ... w3
s
sf(kJ
exp(ijiI
5r ($)),
(A.4)
where D (up, r. , p) is a notation for the path integral D(ra,ro,p)
= Jexp[-+o~ildf
- iV(r)dt]Dr(t),
(-4.5)
f(k) and V(k) are related by d3kf(k) eikSr, V(r) = 1 (W3 s
(A*6)
FEYNMAN’S
PATH INTEGRALS
111
and f(k) = j V(r) eek*’ d3r.
(A-7)
Taking into account convergence arguments it can be proved that interchanging the path-integral sign and the other operator signs is a legitimate manipulation. Define I. as I,, = j exp 1-3 i P dt f !.&(t) r(t) dll Dr (t),
64.8)
fn(f) = l;iI kj 6 (2 - t,).
(A-9)
with
This integral is a gaussian one and can be calculated using the method suggested by Feynman, resulting in 1 -- 1 (q - r. - i i Ic,~~)~+ irO i kJ -j=L j=l 2/3 In - (27$3)+ exp I
(A.lO) with Ti,, defined as
We now aim at calculating
W(ro,B>= jd3r~~(r~,ro,L06).
(A.13
The manipulations to be performed are elementary because of the uniform convergence of the series development for D (r@, ro, /?) holding for every rp. One obtains B B
x exp [-+(k:T,,,
+ *** + 2kn-lknTn-~,n)3.
(A.13)
112
M. J. GOOVAERTS AND J. T. DEVREESE
Calculating obtained
the Laplace transform
...
s
of both members the following result is
J2)
d3knf(kn
explib ’ ckl +
‘*’
+
‘“)I.
s (s + k:) -*a (s + k,Z)
(A.14)
For Re (s) > 0 one obtains the following inequality d3k, If(kl J2)] 1”.
(A.15)
s
Taking into account only s values satisfying Re (s) > 0 and
IsI > j d3k, If(b &)I it
(A.16)
3
follows that is finite.
The same statement respect to s
(A.17)
can easily be proved for the derivations
is finite
i=
of (A.14) with
1,2,....
So as soon as Re (s) is sufficiently large Y [ W(r,, /I), s] is an analytic function of s and W (r. , #I) has to be finite except for an infinite or finite number of discrete J!?values.
REFERENCES 1) Feynman, R.P., Phys. Rev. 91 (1953) 1291. 2) Feynman, R. P., Phys. Rev. 97 (1955) 669. 3) Feynman, R.P., Hellwarth, R. W., Iddings, C. K. and Platzman, P.M., Phys. Rev. 127 (1962) 1004. 4) Platzman, P.M., Phys. Rev. 125 (1962) 1961. 5) Gutzwiller, M., J. math. Phys. 8 (1967) 1979. 6) Goovaerts, M. and Devreese, J., J. math. Phys., to be published. 7) Schulman, L., Phys. Rev. 176 (1968) 1558.
FEYNMAN’S 8) 9) 10) 11)
PATH INTEGRALS
113
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