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Derivation of Feynman's path integral theory based on stochastic mechanics
Volume 137, number 9
PHYSICS LETTERS A
5 June 1989
DERIVATION OF FEYNMAN’S PATH INTEGRAL THEORY BASED ON STOCHASTIC MECHANICS M.S.WANG Department ofPhysics, National Central University, Chung-Li, Taiwan 32054, ROC Received 26 January 1989; accepted for publication 6 April 1989 Communicated by J.P. Vigier
We show that Feynman’s path integral theory can be derived in the framework of stochastic mechanics. The probability of a particle moving from a space—time point (x 0, t0) to another space—time point (x, t) is the stochastic mechanics transition probability instead of the absolute square of the kernel as proposed by Feynman.
Stochastic mechanics [1—5] is a description of quantum phenomena in classical probabilistic terms. It is based on two hypotheses: (1) Every particle of mass m is subject to a Brownian motion with diffusion coefficient /l/2m and no friction. (2) The influence ofan external field on the particle obeys Nelson’s modified version of Newton’s second law. As a consequence of these two hypotheses, the motion of a particle is determined by the Schrodinger equation. On the other hand Feynman’s path integral theory [6] for a nonrelativistic particle also leads to the Schrodinger equation. The path integral theory is based on two hypotheses: (1) The knowledge of a particle occupying a space—time point (x, 1) can only be determined probabilistically by a wave function ~(x, t) such that the probability density is p(x, t) = ~(x, t) 12 (2) The probability of a particle moving from a space—time point (x0, t0) to another space—time point (x, 1) is determined by a kernel K(x, tIx0, t~)such that ~(x,t) =
$
dx0 K(x, 11x0, t0)W(xo,
t~)
(1)
Although both theories are constructed to describe quantum phenomena and they both lead to the Schrodinger equation, the relation between the two theories is unclear. Our recent works [7,8] on the stochastic mechanics transition probability suggest that a relation between the two theories may be established. In this work we show that the two hypotheses of the path integral theory can be derived based on stochastic mechanics and consequently Feynman’s path integral theory is embedded in the framework of stochastic mechanics. First, we shall show that hypothesis (1) of the path integral theory follows from stochastic mechanics. According to stochastic mechanics, the motion of a particle is determined by a forward and a backward stochastic differential equation [2,3]. This requires the knowledge of the forward and backward drift yelocities v~(x, t) and v_ (x, t) which are defined by the probability densityp(x, t) offinding the particle t).
at the space—time point (x,
and the kernel is given by the path integral K(x,11x0 =
$
and a function S(x,
1 ~
t~)
q(to)=xo q(t) =X
t)
~qexp(~dtL(~,q,t)),
(2)
to
(3)
v_(x,t)=~VS(x,t)—~Vln~~). (4) The probability density p(x,
with the classical Lagrangian L (4z, q, 1) = mij~ V(q, ~
—
t)
0375-960l/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
t)
and the function S(x,
satisfy the two coupled differential equations 437
Volume 137, number 9
2m
9t
h2 ~
PHYSICS LETTERS A
(5)
___
1 m
~+_(V~/~).(VS)+~fV2S=0.
5 June 1989
P(x,tIx 0,t0)=
1 2m~
(6)
J
X
q(to)=xo q(t)=x
If we define a function w(x, t) such that ~J(x,t)= \/~~7)exp[(i/h)S(x,t)J,
~qexp{_~ 2mv.)1dt[~m~2_V 2 ~v2~]~ ~
+2mv2 (i+
4m2v2)
h
(7)
then eqs. (5) and (6) will yield the SchrOdinger equation ih~=_~V2W+V~. 8t 2m
\/~~,t)exp[S(x,t)/2mv] ~‘~~0,t0)exp[S(x0,t0)/2mv]
(8)
(11)
The probability density j~(x,t) of this Markov process at the space—time point (x, t) is related to its initial distribution j~(x 0,t0) through the relation fl(x,t)= J~voP(x~tIxo,to)fl(xo,to).
(12)
Choosing the diffusion coefficient v = ih/2m, we have This means that, in the framework of stochastic mechanics, the probability density of finding a particle at a space—time point (x, t) is determined by the wave function ~(x, t) of the Schrodinger equation withp(x, t) = I ~t’(x,t) 2 Thus the hypothesis (1) of the path integral theory is obtained. Next, we shall show that the wave function çt’(x, t) obeys the rule defined by eqs. (1) and (2). Consider a Markov process defined by the Fokker—Planck equation tIx0, t0)+V~[v+(x, t)P(x, tIx0, —
vV~P(x,1Ix0, t0)~o0 ‘
F(x, I I x0, t~)=
W*(x,t)
wx0, t~)K(x, tIxo, t~) (13) Here K(x, t I x0, t~)is precisely the path integral defined in eq. (2) and ~t/~(x, t) is the complex conjugate ofthe wave function ii’(x, t) used to define the Markov process. Defining the function ~p(x, 1) as .
~(x,t) ~i(x, 1) = ~u* (x, 1)
J
t0)]
(9)
where v is the diffusion coefficient, P(x, tixo, t0) is the transition probability, and the drift velocity v~(x, t) is defined by the solutions of eqs. (5) and (6) or equivalently the wave function ~i(x, I) ofthe Schrödinger equation 1VS(x,t)+2v1n~~~). (10) v÷(x,t)= m The initial condition of eq. (9) is
= ~(x, I) ~0P(x, tlx0, t0)fl(x0, ta), (14) and makinguse of eqs. (5), (6), (9) and (10) with v_—ih/2m, it is straightforward to show that ~i(x, I) satisfies the Schrodinger equation
=
~
—
-~-.~-
at
v~+ v~.
2m Thus if the initial condition of ~i(x, t) is chosen to be W(Xo,to)=W(Xo,to), then fl(x 0, t~)=p(x0, t~)
limP(x, tlx0, t0)=ö(x—x0)
çti(x,t)=çii(x,t)
1—. 10
~(x, It can be shown that the solution of eq. (9) subject to the above condition can be expressed in terms of a functional integral [81
=
~
t)
J
&0P(x, tIxo, t0)fl(x0, t0)
K(x, tx0, t0)W(x0, t0)
.
(15)
Thus, the hypothesis (2) of the path integral theory 438
Volume 137, number 9
PHYSICS LETTERS A
5 June 1989
is derived. Although the Schrädinger equation is used
of a particle moving from a space—time point (x0,
in the derivation, it is derived within stochastic mechanics. In summary we have shown that, in the framework of stochastic mechanics, the motion of a particle is entirely determined by a wave function ~(x, 1) which obeys the rule defined by eqs. (1) and (2) and I ~ii(x, t) 12 is the probability density of finding the particle at the space—time point (x, I). Consequently Feynman’s path integral theory is embedded in the framework of stochastic mechanics, As a final remark we note that the choice of ~‘ = 1~/ 2m for the Markov process defined by eq. (9) is simply a mathematical device used to derive eqs. (13) and (15). It should not be confused with the stochastic mechanics diffusion coefficient h/2m. In the framework of stochastic mechanics, the probability
to another space—time point (x, t) is the stochastic mechanics transition probability obtained in eq. (11) with the diffusion coefficient v h/2m instead of I K(x, t I x0, t0) 2 as proposed by Feynman [6].
t0)
References [1] D. Kershaw, Phys. Rev. 136 (1964) B1850. [2] E. Nelson, Phys. Rev. 150 (1966) 1079; Dynamical theories of Brownjan motion (Princeton, 1967); Quantum fluctuations (Princeton, 1985). [3] F. Guerra, Phys. Rep. 77 (1981) 263. [4] F. Guerra and L. Morato, Phys. Rev. D 27 (1983) 1774. [5]R.Marra,Phys.Rev.D36(l987) 1724.
[61R.P. Feynman andA.R. Hibbs, Quantum mechanics andpath integrals (McGraw-Hill, New York, 1965). [7] M.S. Wang, Phys. Rev. A 37 (1988) 1036. [81M.S. Wang, Phys. Rev. A 38 (1988)5401.
439
PHYSICS LETTERS A
5 June 1989
DERIVATION OF FEYNMAN’S PATH INTEGRAL THEORY BASED ON STOCHASTIC MECHANICS M.S.WANG Department ofPhysics, National Central University, Chung-Li, Taiwan 32054, ROC Received 26 January 1989; accepted for publication 6 April 1989 Communicated by J.P. Vigier
We show that Feynman’s path integral theory can be derived in the framework of stochastic mechanics. The probability of a particle moving from a space—time point (x 0, t0) to another space—time point (x, t) is the stochastic mechanics transition probability instead of the absolute square of the kernel as proposed by Feynman.
Stochastic mechanics [1—5] is a description of quantum phenomena in classical probabilistic terms. It is based on two hypotheses: (1) Every particle of mass m is subject to a Brownian motion with diffusion coefficient /l/2m and no friction. (2) The influence ofan external field on the particle obeys Nelson’s modified version of Newton’s second law. As a consequence of these two hypotheses, the motion of a particle is determined by the Schrodinger equation. On the other hand Feynman’s path integral theory [6] for a nonrelativistic particle also leads to the Schrodinger equation. The path integral theory is based on two hypotheses: (1) The knowledge of a particle occupying a space—time point (x, 1) can only be determined probabilistically by a wave function ~(x, t) such that the probability density is p(x, t) = ~(x, t) 12 (2) The probability of a particle moving from a space—time point (x0, t0) to another space—time point (x, 1) is determined by a kernel K(x, tIx0, t~)such that ~(x,t) =
$
dx0 K(x, 11x0, t0)W(xo,
t~)
(1)
Although both theories are constructed to describe quantum phenomena and they both lead to the Schrodinger equation, the relation between the two theories is unclear. Our recent works [7,8] on the stochastic mechanics transition probability suggest that a relation between the two theories may be established. In this work we show that the two hypotheses of the path integral theory can be derived based on stochastic mechanics and consequently Feynman’s path integral theory is embedded in the framework of stochastic mechanics. First, we shall show that hypothesis (1) of the path integral theory follows from stochastic mechanics. According to stochastic mechanics, the motion of a particle is determined by a forward and a backward stochastic differential equation [2,3]. This requires the knowledge of the forward and backward drift yelocities v~(x, t) and v_ (x, t) which are defined by the probability densityp(x, t) offinding the particle t).
at the space—time point (x,
and the kernel is given by the path integral K(x,11x0 =
$
and a function S(x,
1 ~
t~)
q(to)=xo q(t) =X
t)
~qexp(~dtL(~,q,t)),
(2)
to
(3)
v_(x,t)=~VS(x,t)—~Vln~~). (4) The probability density p(x,
with the classical Lagrangian L (4z, q, 1) = mij~ V(q, ~
—
t)
0375-960l/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
t)
and the function S(x,
satisfy the two coupled differential equations 437
Volume 137, number 9
2m
9t
h2 ~
PHYSICS LETTERS A
(5)
___
1 m
~+_(V~/~).(VS)+~fV2S=0.
5 June 1989
P(x,tIx 0,t0)=
1 2m~
(6)
J
X
q(to)=xo q(t)=x
If we define a function w(x, t) such that ~J(x,t)= \/~~7)exp[(i/h)S(x,t)J,
~qexp{_~ 2mv.)1dt[~m~2_V 2 ~v2~]~ ~
+2mv2 (i+
4m2v2)
h
(7)
then eqs. (5) and (6) will yield the SchrOdinger equation ih~=_~V2W+V~. 8t 2m
\/~~,t)exp[S(x,t)/2mv] ~‘~~0,t0)exp[S(x0,t0)/2mv]
(8)
(11)
The probability density j~(x,t) of this Markov process at the space—time point (x, t) is related to its initial distribution j~(x 0,t0) through the relation fl(x,t)= J~voP(x~tIxo,to)fl(xo,to).
(12)
Choosing the diffusion coefficient v = ih/2m, we have This means that, in the framework of stochastic mechanics, the probability density of finding a particle at a space—time point (x, t) is determined by the wave function ~(x, t) of the Schrodinger equation withp(x, t) = I ~t’(x,t) 2 Thus the hypothesis (1) of the path integral theory is obtained. Next, we shall show that the wave function çt’(x, t) obeys the rule defined by eqs. (1) and (2). Consider a Markov process defined by the Fokker—Planck equation tIx0, t0)+V~[v+(x, t)P(x, tIx0, —
vV~P(x,1Ix0, t0)~o0 ‘
F(x, I I x0, t~)=
W*(x,t)
wx0, t~)K(x, tIxo, t~) (13) Here K(x, t I x0, t~)is precisely the path integral defined in eq. (2) and ~t/~(x, t) is the complex conjugate ofthe wave function ii’(x, t) used to define the Markov process. Defining the function ~p(x, 1) as .
~(x,t) ~i(x, 1) = ~u* (x, 1)
J
t0)]
(9)
where v is the diffusion coefficient, P(x, tixo, t0) is the transition probability, and the drift velocity v~(x, t) is defined by the solutions of eqs. (5) and (6) or equivalently the wave function ~i(x, I) ofthe Schrödinger equation 1VS(x,t)+2v1n~~~). (10) v÷(x,t)= m The initial condition of eq. (9) is
= ~(x, I) ~0P(x, tlx0, t0)fl(x0, ta), (14) and makinguse of eqs. (5), (6), (9) and (10) with v_—ih/2m, it is straightforward to show that ~i(x, I) satisfies the Schrodinger equation
=
~
—
-~-.~-
at
v~+ v~.
2m Thus if the initial condition of ~i(x, t) is chosen to be W(Xo,to)=W(Xo,to), then fl(x 0, t~)=p(x0, t~)
limP(x, tlx0, t0)=ö(x—x0)
çti(x,t)=çii(x,t)
1—. 10
~(x, It can be shown that the solution of eq. (9) subject to the above condition can be expressed in terms of a functional integral [81
=
~
t)
J
&0P(x, tIxo, t0)fl(x0, t0)
K(x, tx0, t0)W(x0, t0)
.
(15)
Thus, the hypothesis (2) of the path integral theory 438
Volume 137, number 9
PHYSICS LETTERS A
5 June 1989
is derived. Although the Schrädinger equation is used
of a particle moving from a space—time point (x0,
in the derivation, it is derived within stochastic mechanics. In summary we have shown that, in the framework of stochastic mechanics, the motion of a particle is entirely determined by a wave function ~(x, 1) which obeys the rule defined by eqs. (1) and (2) and I ~ii(x, t) 12 is the probability density of finding the particle at the space—time point (x, I). Consequently Feynman’s path integral theory is embedded in the framework of stochastic mechanics, As a final remark we note that the choice of ~‘ = 1~/ 2m for the Markov process defined by eq. (9) is simply a mathematical device used to derive eqs. (13) and (15). It should not be confused with the stochastic mechanics diffusion coefficient h/2m. In the framework of stochastic mechanics, the probability
to another space—time point (x, t) is the stochastic mechanics transition probability obtained in eq. (11) with the diffusion coefficient v h/2m instead of I K(x, t I x0, t0) 2 as proposed by Feynman [6].
t0)
References [1] D. Kershaw, Phys. Rev. 136 (1964) B1850. [2] E. Nelson, Phys. Rev. 150 (1966) 1079; Dynamical theories of Brownjan motion (Princeton, 1967); Quantum fluctuations (Princeton, 1985). [3] F. Guerra, Phys. Rep. 77 (1981) 263. [4] F. Guerra and L. Morato, Phys. Rev. D 27 (1983) 1774. [5]R.Marra,Phys.Rev.D36(l987) 1724.
[61R.P. Feynman andA.R. Hibbs, Quantum mechanics andpath integrals (McGraw-Hill, New York, 1965). [7] M.S. Wang, Phys. Rev. A 37 (1988) 1036. [81M.S. Wang, Phys. Rev. A 38 (1988)5401.
439