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Some remarks on Nelson's stochastic field
Volume 79A, number 1
PHYSICS LETTERS
15 September 1980
SOME REMARKS ON NELSON’S STOCHASTIC FIELD S.C. LIM Department of Physics, Universiti Kebangsaan Malaysia, Kuala Lumpur, Malaysia Received 26 June 1980
An attempt to extend Nelson’s stochastic quantization procedure to tensor fields indicates that the result of Guerra on the connection between a euclidean Markov scalar field and a stochastic scalar field fails to hold for tensor fields.
Ct
at.
Guerra et a!. [1] have obtained the nice but somewhat unexpected result that a free scalar euclidean Markov field can be interpreted as the ground state of a classical Klein—Gordon field through Nelson’s stochastic quantization procedure [2]. In this way a scalar euclidean Markov field can be considered as defined on the physical Minkowski space—time and not on the fictitious euclidean space—imaginary time as previously assumed. We shall show here by using an elementary calculation that the generalization of this result to free massive tensor fields of arbitrary integer spin fails to hold. For the purpose of comparison later on we shall first give the construction of a class of euclidean Markov tensor fields. Let 0(x) be the real tensor field in Minkowski space—time for massive spin-s bosons, having a local lagrangian density of the Takahashi—Umezawa type [3]: L
=
where A(s) is a local differential (matrix) operator * Denote by W the two-point Wightman function of the 1(p). The two-point Schwinger function is given by tensor field, with the Fourier transform ~V(p)= ?~ S(x—x’)AW[x_x’,i(x 0_x~)] ~.
where x, x’ denote euclidean four-vectors and Ais the matrix transformation required to change the Minkowski metric in W to the euclidean metric in S, and also to preserve the tracelessness of the two-point function. The two-point Schwinger function so obtained is positive semidefinite (for further details refer to Lim [4]). One can then define a euclidean—gaussian tensor field ~f)as the generalized random field with mean zero and covariance S..It is easy to see that 4 is markovian since 1(p) [Afi~(p)J~ = ~(p)A~ =7c(~)A—’, ~ where ~(p) is a polynomial in p and A~1exists for A is a non-singular constant matrix. Nelson’s proof for the scalar case applies [5]. The stochastic quantization of a classical Klein—Gordon tensor field can be carried out in terms of the basic stochastic oscillator process [1] Let .
~11Ml...~S(x,t), xeR3,tER1,
be a real classical tensor field, which satisfies in addition to the Klein—Gordon equation all the subsidiary condi*1
We have suppressed the tensor indices whenever this can be done without ambiguities. The Minkowski metric diag(+1, —1,—i, —1).
is taken as
13
Volume 79A, number 1
PHYSICS LETTERS
15 September 1980
tions (symmetric, traceless and divergenceless). The stochastic tensor field can be Fourier-decomposed in terms of stochastic oscillators as follows: t) = (2i)312
~
~fd3p
e~ 1~(p)[a~(p, t) exp(ip-x) +a~(p,t) exp(—ip.x)]
where e~1..~5(P)is the polarization tensor and a, a* are related to the stochastic oscillator process q by 2[a~(p t) + a~ (p, t)] ~ t) = —2 l/2~ [a~(p, t) a~(p,t)] qx(p, t) = 2 “ with ,
E[q~(p, t)q~’(p’,t’)]
and w =
(I p
2 +
=
—
(2w)~~~(p—p’) exp(—wIt
—
m 2)1 / 2 It is straightforward to compute the covariance:
E[’I’~ 1~(x,t)’1’~1~(x’, t’)]
~
=
(2ir)~
fd3pe~1 ~ (p)e~1 v5(p)
=
(2~~ d~p~
f
.~‘v’...vs
p +m
~
—x’)} exp(—wlt
~
exp[i ~p1(x1
—
t’I)
—
i-i
where O,~~ ~ (p) = ~ ,2(p)e~’1...~5(p)is the spin-s projection operator when p is restricted to the mass-shell [6] A direct comparison between the covariances of a euclidean Markov tensor field and a stochastic tensor field of the same rank shows that they are not the same. We shall illustrate the differences with vector and rank-two tensor fields. The covariance of a euclidean vector field is SJk(x
—
x’) = (2ir)4
f d’1p
2p/pk
~jk+m2
2
exp[i
~p
1(x1
p +m
—
j-1
which differs from the covariance of a stochastic vector field, S~(x—x’, t
—
t’)
(2~~fd4p
~
+ m~pgpp exp[i
p +m
~p1(x1 —xi)]. j1
These two covariances coincide if the stochastic vector field is obtained by the stochastic quantization of a classical Klein—Gordon four-vector field ‘I’1(x, t), with ~114 = i’I’0. Note that the replacement ~‘oby ‘I’~is implicitly contamed in the A-matrix transformation required for obtaining the two-point Schwinger function. Now a new problem arises due to the fact that a Markov vector field with covariance S/k is not divergenceless. In fact 2a/ak)S(x —x’) E[~a/(I~J(x)~ =
ak~k(x)]
~
—
m
(~/m2)(—z~ + m2)S(x —x’) = (i.Vm2)~(x—x’)
where z~is the laplacian operator in R4 and S(x 14
—
x’) is the two-point Schwinger function for a scalar field.
Volume 79A, number 1
PHYSICS LETTERS
15 September 1980
becomes divergenceless under the map
where cJ~is the Fourier transform of ‘T~.However, such a map is non-local in time and does not preserve the Markov property. The transverse vector field so obtained satisfies a weaker property T-positivity, so that it is markovian only with respect to special half-spaces [7]. The proof is similar to the case of a massless vector field [8]. We remark that the loss of one degree of freedom by imposing the divergenceless condition on the euclidean vector field has resulted in making the boundary Hilbert space too small to make the interior independent of the exterior, hence the absence of the Markov property for the transverse vector field. Although the weaker Tpositivity (together with other regularity conditions) is sufficient to recover the relativistic field from the euclidean one, it is not sufficient for Nelson’s stochastic quantization to be carried out because the latter procedure requires the full Markov property. However, a recent result of Okabe [9] shows that a diffusion process satisfying T-positivity can be obtained as a solution of a Langevin-type stochastic differential equation. Thus one may hope to modify Nelson’s stochastic quantization by replacing Ito’s stochastic differential equation with Okabe’s stochastic differential equation. In addition to the above remarks, the situation for stochastic tensor fields is further complicated by the tracelessness condition. We shall consider the stochastic quantization of a classical Klein—Gordon rank-two tensor field —
‘I1/k(X, t),
‘P14
=
E [‘I’Jk(x,t)’Pin(X’, where dlk
=
~t~2
i’P10. The covariance of ‘P/k turns out to be
4 fd4p
t’)]
=
(2ir)
~ iL
dlkdln) exp
p+m
[~ ~
pj~xj x)], —
I-
~/k + m2plpk. This covariance is not traceless (hence not positive-semidefinite) due to the term
—i~~Jk~lflassoc~t~d
with the trace. This is also one of the reasons for introducing the matrix transformationA (containing terms of the form ~ 6J~glU~) to obtain a traceless and positive-semidefinite two-point Schwinger function [4]. From the above discussion we conclude that a euclidean Markov tensor field for massive bosons differs from the ground state of a classical Klein—Gordon tensor field through Nelson’s stochastic quantization. The result of Guerra et al. for a massive scalar field isjust a nice coincidence [10] Similar remarks apply to massless tensor fields, where additional complications arise due to gauge ambiguities [11]. .
References [1] F. Guerra and P. Ruggiero, Phys. Rev. Lett. 31(1973)1022. [2] E. Nelson, Dynamical theories of brownian motion (Princeton U.P., New Jersey, 1967); E. Nelson, Phys. Rev. 150 (1966) 1079. [3] U. Umezawa and Y. Takahashi, Prog. Theor. Phys. 9(1951)1; Y. Takahashi, An introduction to field quantization (Pergamon, Oxford, 1969). [4] S.C. Lim, Phys. Lett. 77B (1978) 287; S.C. Lim, J. Phys. A (1980), to be published. [5] E. Nelson, J. Func. Anal. 12(1973) 211. [6] L.P.S. Singh and C.R. Hagen, Phys. Rev. D9 (1974) 898; R.L. Ingraham, Prog. Theor. Phys. 51(1974) 249. [7] G.C. Hegerbeldt, Commun. Math. Phys. 45 (1975) 137. [8] S.C. Lim, J. Phys. A12 (1979) 1899. [9] Y. Okabe, J. Sci. A. Kyoto Univ. 26(1979)115. [10] F. Guerra, Private communications. Professor Guerra has indicated that his result on stochastic scalar field holds for the higher spin case, but we have not seen this generalization published. [11] S.C. Lim, in preparation.
15
PHYSICS LETTERS
15 September 1980
SOME REMARKS ON NELSON’S STOCHASTIC FIELD S.C. LIM Department of Physics, Universiti Kebangsaan Malaysia, Kuala Lumpur, Malaysia Received 26 June 1980
An attempt to extend Nelson’s stochastic quantization procedure to tensor fields indicates that the result of Guerra on the connection between a euclidean Markov scalar field and a stochastic scalar field fails to hold for tensor fields.
Ct
at.
Guerra et a!. [1] have obtained the nice but somewhat unexpected result that a free scalar euclidean Markov field can be interpreted as the ground state of a classical Klein—Gordon field through Nelson’s stochastic quantization procedure [2]. In this way a scalar euclidean Markov field can be considered as defined on the physical Minkowski space—time and not on the fictitious euclidean space—imaginary time as previously assumed. We shall show here by using an elementary calculation that the generalization of this result to free massive tensor fields of arbitrary integer spin fails to hold. For the purpose of comparison later on we shall first give the construction of a class of euclidean Markov tensor fields. Let 0(x) be the real tensor field in Minkowski space—time for massive spin-s bosons, having a local lagrangian density of the Takahashi—Umezawa type [3]: L
=
where A(s) is a local differential (matrix) operator * Denote by W the two-point Wightman function of the 1(p). The two-point Schwinger function is given by tensor field, with the Fourier transform ~V(p)= ?~ S(x—x’)AW[x_x’,i(x 0_x~)] ~.
where x, x’ denote euclidean four-vectors and Ais the matrix transformation required to change the Minkowski metric in W to the euclidean metric in S, and also to preserve the tracelessness of the two-point function. The two-point Schwinger function so obtained is positive semidefinite (for further details refer to Lim [4]). One can then define a euclidean—gaussian tensor field ~f)as the generalized random field with mean zero and covariance S..It is easy to see that 4 is markovian since 1(p) [Afi~(p)J~ = ~(p)A~ =7c(~)A—’, ~ where ~(p) is a polynomial in p and A~1exists for A is a non-singular constant matrix. Nelson’s proof for the scalar case applies [5]. The stochastic quantization of a classical Klein—Gordon tensor field can be carried out in terms of the basic stochastic oscillator process [1] Let .
~11Ml...~S(x,t), xeR3,tER1,
be a real classical tensor field, which satisfies in addition to the Klein—Gordon equation all the subsidiary condi*1
We have suppressed the tensor indices whenever this can be done without ambiguities. The Minkowski metric diag(+1, —1,—i, —1).
is taken as
13
Volume 79A, number 1
PHYSICS LETTERS
15 September 1980
tions (symmetric, traceless and divergenceless). The stochastic tensor field can be Fourier-decomposed in terms of stochastic oscillators as follows: t) = (2i)312
~
~fd3p
e~ 1~(p)[a~(p, t) exp(ip-x) +a~(p,t) exp(—ip.x)]
where e~1..~5(P)is the polarization tensor and a, a* are related to the stochastic oscillator process q by 2[a~(p t) + a~ (p, t)] ~ t) = —2 l/2~ [a~(p, t) a~(p,t)] qx(p, t) = 2 “ with ,
E[q~(p, t)q~’(p’,t’)]
and w =
(I p
2 +
=
—
(2w)~~~(p—p’) exp(—wIt
—
m 2)1 / 2 It is straightforward to compute the covariance:
E[’I’~ 1~(x,t)’1’~1~(x’, t’)]
~
=
(2ir)~
fd3pe~1 ~ (p)e~1 v5(p)
=
(2~~ d~p~
f
.~‘v’...vs
p +m
~
—x’)} exp(—wlt
~
exp[i ~p1(x1
—
t’I)
—
i-i
where O,~~ ~ (p) = ~ ,2(p)e~’1...~5(p)is the spin-s projection operator when p is restricted to the mass-shell [6] A direct comparison between the covariances of a euclidean Markov tensor field and a stochastic tensor field of the same rank shows that they are not the same. We shall illustrate the differences with vector and rank-two tensor fields. The covariance of a euclidean vector field is SJk(x
—
x’) = (2ir)4
f d’1p
2p/pk
~jk+m2
2
exp[i
~p
1(x1
p +m
—
j-1
which differs from the covariance of a stochastic vector field, S~(x—x’, t
—
t’)
(2~~fd4p
~
+ m~pgpp exp[i
p +m
~p1(x1 —xi)]. j1
These two covariances coincide if the stochastic vector field is obtained by the stochastic quantization of a classical Klein—Gordon four-vector field ‘I’1(x, t), with ~114 = i’I’0. Note that the replacement ~‘oby ‘I’~is implicitly contamed in the A-matrix transformation required for obtaining the two-point Schwinger function. Now a new problem arises due to the fact that a Markov vector field with covariance S/k is not divergenceless. In fact 2a/ak)S(x —x’) E[~a/(I~J(x)~ =
ak~k(x)]
~
—
m
(~/m2)(—z~ + m2)S(x —x’) = (i.Vm2)~(x—x’)
where z~is the laplacian operator in R4 and S(x 14
—
x’) is the two-point Schwinger function for a scalar field.
Volume 79A, number 1
PHYSICS LETTERS
15 September 1980
becomes divergenceless under the map
where cJ~is the Fourier transform of ‘T~.However, such a map is non-local in time and does not preserve the Markov property. The transverse vector field so obtained satisfies a weaker property T-positivity, so that it is markovian only with respect to special half-spaces [7]. The proof is similar to the case of a massless vector field [8]. We remark that the loss of one degree of freedom by imposing the divergenceless condition on the euclidean vector field has resulted in making the boundary Hilbert space too small to make the interior independent of the exterior, hence the absence of the Markov property for the transverse vector field. Although the weaker Tpositivity (together with other regularity conditions) is sufficient to recover the relativistic field from the euclidean one, it is not sufficient for Nelson’s stochastic quantization to be carried out because the latter procedure requires the full Markov property. However, a recent result of Okabe [9] shows that a diffusion process satisfying T-positivity can be obtained as a solution of a Langevin-type stochastic differential equation. Thus one may hope to modify Nelson’s stochastic quantization by replacing Ito’s stochastic differential equation with Okabe’s stochastic differential equation. In addition to the above remarks, the situation for stochastic tensor fields is further complicated by the tracelessness condition. We shall consider the stochastic quantization of a classical Klein—Gordon rank-two tensor field —
‘I1/k(X, t),
‘P14
=
E [‘I’Jk(x,t)’Pin(X’, where dlk
=
~t~2
i’P10. The covariance of ‘P/k turns out to be
4 fd4p
t’)]
=
(2ir)
~ iL
dlkdln) exp
p+m
[~ ~
pj~xj x)], —
I-
~/k + m2plpk. This covariance is not traceless (hence not positive-semidefinite) due to the term
—i~~Jk~lflassoc~t~d
with the trace. This is also one of the reasons for introducing the matrix transformationA (containing terms of the form ~ 6J~glU~) to obtain a traceless and positive-semidefinite two-point Schwinger function [4]. From the above discussion we conclude that a euclidean Markov tensor field for massive bosons differs from the ground state of a classical Klein—Gordon tensor field through Nelson’s stochastic quantization. The result of Guerra et al. for a massive scalar field isjust a nice coincidence [10] Similar remarks apply to massless tensor fields, where additional complications arise due to gauge ambiguities [11]. .
References [1] F. Guerra and P. Ruggiero, Phys. Rev. Lett. 31(1973)1022. [2] E. Nelson, Dynamical theories of brownian motion (Princeton U.P., New Jersey, 1967); E. Nelson, Phys. Rev. 150 (1966) 1079. [3] U. Umezawa and Y. Takahashi, Prog. Theor. Phys. 9(1951)1; Y. Takahashi, An introduction to field quantization (Pergamon, Oxford, 1969). [4] S.C. Lim, Phys. Lett. 77B (1978) 287; S.C. Lim, J. Phys. A (1980), to be published. [5] E. Nelson, J. Func. Anal. 12(1973) 211. [6] L.P.S. Singh and C.R. Hagen, Phys. Rev. D9 (1974) 898; R.L. Ingraham, Prog. Theor. Phys. 51(1974) 249. [7] G.C. Hegerbeldt, Commun. Math. Phys. 45 (1975) 137. [8] S.C. Lim, J. Phys. A12 (1979) 1899. [9] Y. Okabe, J. Sci. A. Kyoto Univ. 26(1979)115. [10] F. Guerra, Private communications. Professor Guerra has indicated that his result on stochastic scalar field holds for the higher spin case, but we have not seen this generalization published. [11] S.C. Lim, in preparation.
15