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Path integrals in Riemannian spaces
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
PATH INTEGRALS IN RIEMANNIAN SPACES H. DEKKER Physics Laboratory TNO, The Hague, The Netherlands
Received 7 November 1979
The covariant path integral for a free particle in curved space will be evaluated by means of a spectral analysis of smooth paths. No discretization rule will be required to put the action on a lattice. The connection between the resulting quantum hamiltonian and the Onsager—Machlup lagrangian for diffusion processes will be discussed. The present treatment corrects an earlier version.
1. Introduction. Recently the Fourier series analysis of continuous and differentiable trajectories, which was successfully employed in flat spaces [1], has been applied to the evaluation of path integrals in riemanalan geometries [2,3] in an attempt to be conclusive in the debate concerning the precise fraction of the curvature scalar potential R that must be added to the flat space Onsager—Macblup lagrangian for diffusion processes [4—7]or, which is the equivalent problem, to the flat space hamiltonian of quantum mechanics [8 151 In refs. [2,31 the path integral was first transformed to local euclidean frames, where the spectral analysis was done; the skeletonized action was then transformed back to the globally riemannian coordinates, The result was in agreement with for example refs. [5, 6,8], but differed from refs. [4,7]. It has subsequently been argued [16] that the definitionof the local frames in refs. [2,3] cannot be made precise enough within the path integral if space is essentially curved. In order to overcome this difficulty the spectral applied presently directly in the 2. Evaluation ofthe path integral in Riemann normal coordinates. The covariant path integral for the quantum-mechanical free particle in a riemannian geometry of M dimensions reads (see also refs. [17,18]):
8
K(xf, tflXl, t~)dxf {x(t) } exp [iS({x(t)})]
= J7~CD
(2.1)
,
tf
S({x(t)})
=
f L(x(t),
~(t))
dt
(2.2)
,
.
L(x(t), x(t))
=
~g~~(x)XMXV 2N+1
~
~x(t)} = ~im
~
1
(2.3)
,
~
M —1/2
[(~~~)] 2+
g~ ~k’ (2.4)
where {x(t)} {x(t) = {x~(t)},~uE [0, M 1] }, xk = X(tk), dxk = dx~dx)~..., and where g~~(x) is the covariant metric tensor with determinant gk = g(x~)= I Ig~(x~)I I. It should be noted that eq. (2.4) includes the final point Xf = x2N+ 1,but not the initial one x1 = x0. The transition kernel satisfies the integral relation: —
K(x, tlx”,
~
=
f K(X, tlx’, t’)K(x’, t’Ix”, r”)
(2.5)
from which we will derive the equivalent differential equation, which amounts to the calculation of the short time transition kernel. Since in eq. (2.1) one must probe the stochastic trajectories connecting the fixed end
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
points x1 x(t~)and Xf = X(tf) at an infinity of intermediate times tk E (ti, tf) no matter the value of r = tf t~,these paths should a priori be fully arbitrary even if r tends to very small values. The actual calculation proceeds most elegantly in Riemann nonnal coordinates which to willchoose be denoted byy’1. It will further be [19], convenient the final point to be the origin of both the normal coordinates and time, so that Yf = 0, tf = 0 and t~= —T (see also ref. [13]). The trajectories betweeny~and zero will be written as the sum of their geodesic part, which in normal coordinates by definition is a straight line through the origin, and an arbitrary remainder which wifi be represented as a Fourier series, as follows ~:
eq. (2.7) one separates the action in two obvious parts, ~ and S2. The immediate result for ~o reads:
y~(t)= —yes
S
—
S0
N
+r
~
+ lim [a~ sin 2irns + b~(1— cos 2irns)], N-+°° n1 (2.6)
where s = t/r. By this technique we draw the ultimate mathematical consequences of the important conception of full stochasticity of the paths over any time interval. That is, we do not need to indicate a time lapse on which the process is a priori assumed to be effectively deterministic in the sense that only the geodesic path contributes theare propagation refs. [5,8,13]). In this waytowe also able to(see pute.g. the action on a lattice without reference to any ad-hoc discretization prescription. Inserting eq. (2.6) into eq. (2.1), using the pertinent formula for the expansion of the metric tensor and its determinant (see e.g. refs. [7,19]), 1
=g~~(0) + ~(R~,
~
2n2(al.Lav+ bI2bv)
(0) n=i ~ ~
~
‘~
(2 9)
‘~ ~1
which as usual leads to the calculation of gaussian integrals over the Fourier coefficients. One has (a~a~) = irg’~’(0)~nm12~’T2fl2 = (b~b~), while more complicated moments easily follow by means of the gaussian theorem. Using the above results and the outcome ir2/6 for the sum of the inverse squares of all natural numbers (e.g. refs. [20,21]), one finds: 2
N
=
~
—~rNR0 ~rR0,
(2.10)
—
where ~ denotes equivalence within path integrals, while R has been defined with Weinberg’s sign convention [3,22]. Finally, with eq. (2.8) one evaluates the measure (2.4). It turns out to effectively add a contribution to the action that precisely cancels the singular term in eq. (2.9). Thus, one obtains the short time propagator as: ~
~
\
<>~ 1/2
i~u,uy~,—rj= iig0
r~
exp~i
2.11
,
y~~r ~ rR0 and ~?Z
with cS ~ (1/2r) 2. g~~(0)
—
(2irir)_M/
3. The hamiltonian and the Onsager—Machiup lagrangian. Inserting eq. (2.11) into the integral rela-
tion (2.5)— replacingx byy, in particularx’ byy~ and employing Feynman’s technique of integration over the prepointy~(see e.g. refs. [17,23]), one readily obtains in an obvious notation: —
g~~(y)
0 +R~,~0)0yXya
+...,
g(y) =g(0){1 + ~R~~(0)yxya
+ ...},
(2.7) (2.8)
and evaluating several simple goniometric integrals, it is a rather straightforward and even not too tedious matter to obtain the fmal result. Although further details wifi be reported elsewhere, a few of the constituent intermediate formulae will be given here. With *1 Rather than the previously used pure sine series [1—31 the
present representation is both more complete and leads to more simple algebra. Nonetheless, we have verified that the pure sine series leads to the same result.
2K/ay/23yv) OK0 . (3.1) —iaK0/at = lg4LV(a If, as before [3], we wish to ensure that I V.’ 2 dx, representing a probability in quantum mechanics, transforms like a scalar, we should let K ~,gl/4. Recasting eq. (3.1) into its correct covariant form [24] one then finds the Schrodinger equation as: —
-+
—i
~
at
=
~g1/4
±
gl/4gI2Vgl/4
ax’-~
—p-- ~ aXV
+ 8
The hamiltonian in this Schrodinger equation differs from the one found in ref. where an additional [31,
9
Volume 76A, number 1
PHYSICS LETTERS
was found, but which is now believed to be valid in fact in flat space only [16]. It should be clear from a comparison with ref. [3] that eq. (3.2) implies that a term must be added to the flat space Onsager—Machluplagrangian, which thus will read: L(x, ~)
&) +.~g’/2(a/ax~)g’/2aM+~R, =
~g(k’~
—
aI2)(~v—
(3.3)
where o~12(x)is the contravariant drift vector of the pertinent diffusion process. The ~R, obtained from the evaluation of the path integral by means of the spectral analysis of smooth paths in riemannian coordinates, contrasts with the ~R and ~R floating around in the literature. Presumably these are due to an intrinsically flat space or a priori geodesic path evaluation. However, the present result (3.1) completely agrees *2 with that obtained by Kawai [25,26] in extending Schwinger’s quantum action principle [271 to curved spaces. This agreement is deemed to be non-accidental. Notice a difference in sign in the definition of the curvature scalar.
*2
References [1] H. Dekker, Phys. Lett. 68A (1978) 137. [2] H. Dekker, Phys. Lett. 69A (1978) 241. [3] H. Dekker, Phys. Rev. A19 (1979) 2102.
10
3 March 1980
[4]R. Graham, Z. Phys. B26 (1977) 281. [5] W. Weiss, Z. Phys. B30 (1978) 429. F. Langouche, D.(Louvain). Roekaerts and E. Tirapegui, preprint KUL-TF-78/027 [7] U. Deininghaus and R. Graham, Z. Phys. B34 (1979) 211. [81B.S. DeWitt, Rev. Mod. Phys. 29 (1957) 377. [9] KS. Cheng, J. Math. Phys. 13 (1972) 1723. [10] S.I. Ben-Abraham and A. Lonke, J. Math. Phys. 14 (1973) 1935. [11] 5.5. Dowker, 5. Phys. A7 (1974) 1256. [12] M.M. Mizrahi, I. Math. Phys. 16 (1975) 2201. [13] G.A. Ringwood, J. Phys. A9 (1976) 1253. [14] T. Kimura, Prog. Theor. Phys. 58(1977)1964. [15] A.C. Hirschfeld, Phys. Lett. 67A (1978) 5. [16] F. Langouche, D.(Louvain). Roekaerts and E. Tirapegui, preprint KUL-TF-79/018 [17] R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). [18] R.P. Feynman, Statistical mechanics (Benjamin, Reading, MA, 1972). [19] 0. Veblen, Invariants of quadratic differential forms (Cambridge U.P., Cambridge, 1952). [20] L.B.W. Jolley, Summation of series (Dover Publ., New York, 1961). [21J1.S. Gradshteynand I.W. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1965). [22] 5. Weinberg, Gravitation and cosmology (Wiley, New York, 1972). [23] H. Dekker, Physica 84A (1976) 205. [24] Rev. 32 (1928) 812. [25] B. H. Podoiski, Kamo andPhys. T. Kawai, Prog. Theor. Phys. 50 (1973) 680. [6]
[26] T. Kawai, Found. Phys. 5 (1975) 143. [27] 5. Schwinger, Quantum kinematics and dynamics (Benjamin, New York, 1970).
PHYSICS LETTERS
3 March 1980
PATH INTEGRALS IN RIEMANNIAN SPACES H. DEKKER Physics Laboratory TNO, The Hague, The Netherlands
Received 7 November 1979
The covariant path integral for a free particle in curved space will be evaluated by means of a spectral analysis of smooth paths. No discretization rule will be required to put the action on a lattice. The connection between the resulting quantum hamiltonian and the Onsager—Machlup lagrangian for diffusion processes will be discussed. The present treatment corrects an earlier version.
1. Introduction. Recently the Fourier series analysis of continuous and differentiable trajectories, which was successfully employed in flat spaces [1], has been applied to the evaluation of path integrals in riemanalan geometries [2,3] in an attempt to be conclusive in the debate concerning the precise fraction of the curvature scalar potential R that must be added to the flat space Onsager—Macblup lagrangian for diffusion processes [4—7]or, which is the equivalent problem, to the flat space hamiltonian of quantum mechanics [8 151 In refs. [2,31 the path integral was first transformed to local euclidean frames, where the spectral analysis was done; the skeletonized action was then transformed back to the globally riemannian coordinates, The result was in agreement with for example refs. [5, 6,8], but differed from refs. [4,7]. It has subsequently been argued [16] that the definitionof the local frames in refs. [2,3] cannot be made precise enough within the path integral if space is essentially curved. In order to overcome this difficulty the spectral applied presently directly in the 2. Evaluation ofthe path integral in Riemann normal coordinates. The covariant path integral for the quantum-mechanical free particle in a riemannian geometry of M dimensions reads (see also refs. [17,18]):
8
K(xf, tflXl, t~)dxf {x(t) } exp [iS({x(t)})]
= J7~CD
(2.1)
,
tf
S({x(t)})
=
f L(x(t),
~(t))
dt
(2.2)
,
.
L(x(t), x(t))
=
~g~~(x)XMXV 2N+1
~
~x(t)} = ~im
~
1
(2.3)
,
~
M —1/2
[(~~~)] 2+
g~ ~k’ (2.4)
where {x(t)} {x(t) = {x~(t)},~uE [0, M 1] }, xk = X(tk), dxk = dx~dx)~..., and where g~~(x) is the covariant metric tensor with determinant gk = g(x~)= I Ig~(x~)I I. It should be noted that eq. (2.4) includes the final point Xf = x2N+ 1,but not the initial one x1 = x0. The transition kernel satisfies the integral relation: —
K(x, tlx”,
~
=
f K(X, tlx’, t’)K(x’, t’Ix”, r”)
(2.5)
from which we will derive the equivalent differential equation, which amounts to the calculation of the short time transition kernel. Since in eq. (2.1) one must probe the stochastic trajectories connecting the fixed end
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
points x1 x(t~)and Xf = X(tf) at an infinity of intermediate times tk E (ti, tf) no matter the value of r = tf t~,these paths should a priori be fully arbitrary even if r tends to very small values. The actual calculation proceeds most elegantly in Riemann nonnal coordinates which to willchoose be denoted byy’1. It will further be [19], convenient the final point to be the origin of both the normal coordinates and time, so that Yf = 0, tf = 0 and t~= —T (see also ref. [13]). The trajectories betweeny~and zero will be written as the sum of their geodesic part, which in normal coordinates by definition is a straight line through the origin, and an arbitrary remainder which wifi be represented as a Fourier series, as follows ~:
eq. (2.7) one separates the action in two obvious parts, ~ and S2. The immediate result for ~o reads:
y~(t)= —yes
S
—
S0
N
+r
~
+ lim [a~ sin 2irns + b~(1— cos 2irns)], N-+°° n1 (2.6)
where s = t/r. By this technique we draw the ultimate mathematical consequences of the important conception of full stochasticity of the paths over any time interval. That is, we do not need to indicate a time lapse on which the process is a priori assumed to be effectively deterministic in the sense that only the geodesic path contributes theare propagation refs. [5,8,13]). In this waytowe also able to(see pute.g. the action on a lattice without reference to any ad-hoc discretization prescription. Inserting eq. (2.6) into eq. (2.1), using the pertinent formula for the expansion of the metric tensor and its determinant (see e.g. refs. [7,19]), 1
=g~~(0) + ~(R~,
~
2n2(al.Lav+ bI2bv)
(0) n=i ~ ~
~
‘~
(2 9)
‘~ ~1
which as usual leads to the calculation of gaussian integrals over the Fourier coefficients. One has (a~a~) = irg’~’(0)~nm12~’T2fl2 = (b~b~), while more complicated moments easily follow by means of the gaussian theorem. Using the above results and the outcome ir2/6 for the sum of the inverse squares of all natural numbers (e.g. refs. [20,21]), one finds: 2
N
=
~
—~rNR0 ~rR0,
(2.10)
—
where ~ denotes equivalence within path integrals, while R has been defined with Weinberg’s sign convention [3,22]. Finally, with eq. (2.8) one evaluates the measure (2.4). It turns out to effectively add a contribution to the action that precisely cancels the singular term in eq. (2.9). Thus, one obtains the short time propagator as: ~
~
\
<>~ 1/2
i~u,uy~,—rj= iig0
r~
exp~i
2.11
,
y~~r ~ rR0 and ~?Z
with cS ~ (1/2r) 2. g~~(0)
—
(2irir)_M/
3. The hamiltonian and the Onsager—Machiup lagrangian. Inserting eq. (2.11) into the integral rela-
tion (2.5)— replacingx byy, in particularx’ byy~ and employing Feynman’s technique of integration over the prepointy~(see e.g. refs. [17,23]), one readily obtains in an obvious notation: —
g~~(y)
0 +R~,~0)0yXya
+...,
g(y) =g(0){1 + ~R~~(0)yxya
+ ...},
(2.7) (2.8)
and evaluating several simple goniometric integrals, it is a rather straightforward and even not too tedious matter to obtain the fmal result. Although further details wifi be reported elsewhere, a few of the constituent intermediate formulae will be given here. With *1 Rather than the previously used pure sine series [1—31 the
present representation is both more complete and leads to more simple algebra. Nonetheless, we have verified that the pure sine series leads to the same result.
2K/ay/23yv) OK0 . (3.1) —iaK0/at = lg4LV(a If, as before [3], we wish to ensure that I V.’ 2 dx, representing a probability in quantum mechanics, transforms like a scalar, we should let K ~,gl/4. Recasting eq. (3.1) into its correct covariant form [24] one then finds the Schrodinger equation as: —
-+
—i
~
at
=
~g1/4
±
gl/4gI2Vgl/4
ax’-~
—p-- ~ aXV
+ 8
The hamiltonian in this Schrodinger equation differs from the one found in ref. where an additional [31,
9
Volume 76A, number 1
PHYSICS LETTERS
was found, but which is now believed to be valid in fact in flat space only [16]. It should be clear from a comparison with ref. [3] that eq. (3.2) implies that a term must be added to the flat space Onsager—Machluplagrangian, which thus will read: L(x, ~)
&) +.~g’/2(a/ax~)g’/2aM+~R, =
~g(k’~
—
aI2)(~v—
(3.3)
where o~12(x)is the contravariant drift vector of the pertinent diffusion process. The ~R, obtained from the evaluation of the path integral by means of the spectral analysis of smooth paths in riemannian coordinates, contrasts with the ~R and ~R floating around in the literature. Presumably these are due to an intrinsically flat space or a priori geodesic path evaluation. However, the present result (3.1) completely agrees *2 with that obtained by Kawai [25,26] in extending Schwinger’s quantum action principle [271 to curved spaces. This agreement is deemed to be non-accidental. Notice a difference in sign in the definition of the curvature scalar.
*2
References [1] H. Dekker, Phys. Lett. 68A (1978) 137. [2] H. Dekker, Phys. Lett. 69A (1978) 241. [3] H. Dekker, Phys. Rev. A19 (1979) 2102.
10
3 March 1980
[4]R. Graham, Z. Phys. B26 (1977) 281. [5] W. Weiss, Z. Phys. B30 (1978) 429. F. Langouche, D.(Louvain). Roekaerts and E. Tirapegui, preprint KUL-TF-78/027 [7] U. Deininghaus and R. Graham, Z. Phys. B34 (1979) 211. [81B.S. DeWitt, Rev. Mod. Phys. 29 (1957) 377. [9] KS. Cheng, J. Math. Phys. 13 (1972) 1723. [10] S.I. Ben-Abraham and A. Lonke, J. Math. Phys. 14 (1973) 1935. [11] 5.5. Dowker, 5. Phys. A7 (1974) 1256. [12] M.M. Mizrahi, I. Math. Phys. 16 (1975) 2201. [13] G.A. Ringwood, J. Phys. A9 (1976) 1253. [14] T. Kimura, Prog. Theor. Phys. 58(1977)1964. [15] A.C. Hirschfeld, Phys. Lett. 67A (1978) 5. [16] F. Langouche, D.(Louvain). Roekaerts and E. Tirapegui, preprint KUL-TF-79/018 [17] R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). [18] R.P. Feynman, Statistical mechanics (Benjamin, Reading, MA, 1972). [19] 0. Veblen, Invariants of quadratic differential forms (Cambridge U.P., Cambridge, 1952). [20] L.B.W. Jolley, Summation of series (Dover Publ., New York, 1961). [21J1.S. Gradshteynand I.W. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1965). [22] 5. Weinberg, Gravitation and cosmology (Wiley, New York, 1972). [23] H. Dekker, Physica 84A (1976) 205. [24] Rev. 32 (1928) 812. [25] B. H. Podoiski, Kamo andPhys. T. Kawai, Prog. Theor. Phys. 50 (1973) 680. [6]
[26] T. Kawai, Found. Phys. 5 (1975) 143. [27] 5. Schwinger, Quantum kinematics and dynamics (Benjamin, New York, 1970).