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Euclidean field theory without reflection positivity. the case of photon field
Vol. 31 (1992)
REPORTS
ON MATHEMATICAL
PHYSICS
No. 2
EUCLIDEAN FIELD THEORY WITHOUT REFLECTION POSITMTY. THE CASE OF PHOTON FIELD
Institute of Theoretical Physics, University of Wrodaw, Wroclaw, Poland (Received September 13, 1991 - Revised November 5, 1991) We study the properties of euclidean field theory corresponding to the free photon field in the covariant (non-positive) gauge.
1. Introduction
In QFT, positivity of the Wightman state W = {Wn} corresponds to the reflectionpositivity of the euclidean Schwinger state S = {Sn} [l]. As it was shown in [2], QFT without Wightman positivity [3] but satisfying so-called Hilbert space structure condition, still allows the euclidean continuation. The corresponding Schwinger state S satisfies euclidean covariance and symmetry, but instead of reflection positivity, S satisfies the following euclidean Hilbert space structure condition (for notation and ail details see [2]): IS(eFY 8 G+)l I PS(F+)PS(G+), where ps is a non-degenerate
Hilbert semi-norm on the Borchers algebra f3+ over S+ (R4) P.s(F+) I IF+ls.
The above condition allows also recovering of Wightman state satisfying Hilbert space structure condition. In this note, we apply the general strategy of [2] to the simple case of photon field in the covariant (non-positive) gauge. In this way we obtain Schwinger functions of the theory which are not reflection-positive. On the other hand they are Nelson-Symanzik positive, hence they are expectation values of some generalized stochastic process. In the paper we examine in details the properties of this process and generalize Nelson reconstruction theory [4] to the case of Euclidean photon field. It seems that the method of reconstruction developed in the paper can be applied only to the free field case. 2. Photon field in Gupta-Bleuler
gauge
The theory is defined by the two-point function (11.1)
180
L. JAKdBCZYK
with f,g E S(R4)04 and I+“(q)
= (-GP’“)@(q),
(11.2)
where F;(q) is the Fourier transform of two-point Wightman-function of massless scalar field. It is easy to see that IV2 is not Wightman positive, but satisfies Hilbert space structure condition IW2(f
8 c?)l 5 PkmPW(9)
(11.3)
with semi-norm pw defined by (11.4) pw is obviously S-continuous
and non-degenerate in_- the 1sense that kerpw = ker IV. Hence, by spectral property, pw is a distribution on S(R+“) and we can define 5(T) = PWW
for j(q) = j(q) t {so 2 0) E S(%+4). Applying the results of [2], we conclude that the corresponding Schwinger function Sz will satisfy euclidean Hilbert space structure condition with semi-norm p.#+) defined in terms of pw(f) by the construction P.df+)
(11.5)
= %1+>7
where f+(<)e- (4°~o-is”~)d
1+(q) = J so
Pdf+) =
(2 J &wG?~ljiri(4J),2)1’2. (11.6)
p=o
On the other hand, we can directly continue Wz to the euclidean points [5]. Let us denote by tS the points in the extended permuted tube Iex’J’ of the form zs = (iz’, T?) with z”, zlc E R (Schwingerpoinfs). Then the corresponding Euclidean points zE are given by XE
with I = diag(i, 1, l,l). in the following way
=
1-‘*S
In the case of scalar field, the Schwinger function Sz is defined S2(X:B,
X,“)
=
W2(IXf,
IX?>,
(11.7)
where Wz is the Wightman function analytic in I extJ’. Since Wz is invariant with respect to all complex Lorentz transformations, Sz is invariant with respect to the transformations L if the form L = I-GlI,
(11.8)
EUCLIDEAN FIELD THEORY WITHOUT REFLECIION POSITIVITY
181
where A E L+(C) is such that A maps Schwinger points onto the Schwinger points. One can check that A E L+(C) satisfying GA = AG has this property, and the corresponding L forms the group SO(4). In the vector case, the following definition W)
(11.9)
guarantees the covariance of S2 under SO(4) if W2 in L+(C)-covariant. case we obtain
Hence, in our
%W&W)
= c (W,,V, “IY”2
s2p,
,p2 ($7
$)
(W,,V,~“,“,V&
= @““2S(&
(11.10)
@)
with s(r) = J e- &0-iT.$)@(q)&_ Now we can discuss the problem of reflection-positivity be the time reflection. In the vector case T is represented (of),@)
= c
(11.11) of $. Let r(z”, Ti?)=(-z”, by the following operator
T’)
(11.12)
qW_fY(~~) ”
with M = diag(-1,
1, l,l).
Using u(r) we have S2(4f-)f+
Hence &(u(r)f+
@ 9+)
= @WC?,
(11.13)
@ 8+).
@ f+) is not positive definite. On the other hand IS2(4T)f+
(11.14)
@ cl+)1 5 ?%G+M9+)
with Pdf+)
= (5
J
d4~d4YS2@ - Yp+,,w+,(y))
1’2
p=o = (2 J p=o
&7m7+du~,‘)
1’2
so we again obtain (11.6). 3. Euclidean field theory and generalized
Nelson construction
Let (~2, C, p) be a probability space. If X is a topological vector space a generalized stochastic process over X is a stochastic process indexed by elements of X, i.e. for f E X A(f) is a random variable on (Q, C, CL). Let A be a generalized stochastic process over S(Rd)@k. If A is an open set in Rd, CA denotes a-algebra generated by A(f) with supp f C A. For an arbitrary subset of Rd we define cA = n A,>A
C,
182
L.JAKi)BCZYK
(where A’ is an open set containing A). If we denote by E{.jC4} conditional expectation with respect to CA, then the Murkovproperty of A means that (111.1)
E{QlCAC)] = E{Ql&4]
for any positive or integrable random variable cy in CA and any open set A c Rd [4]. Let us consider now the Schwinger function & discussed in Section II. By NelsonSymanzik positivity of Sz, (f, g) = &(f 18g) defines the Hilbert space inner product in SJ#Z4)@? Let ‘FIbe the real Hilbert space completion of SR(R4)@4 with respect to (, ). (‘Ii, (, )) defines the unique generalized Gaussian stochastic process A over S(&)@4 [4]. It means that we have probability space (Q, C, p) and the random variables A(f), f E S(&)@4 are Gaussian with covariance -IW.f)A(g)
= s A(f)( R
= s2c.f@ 9).
(111.2)
Using the properties of Sz we can obtain the following properties of euclidean field A: Euclidean covuriance. For every a E R4 and R E SO(4) there exists unitary operator T(a, R) defined on L2(0, C, CL)such that
T(a, R)A(f)T(a,
R)-’ = A&R)).
T(a, R) is given by T(z~(u, R)), w h ere u(u, R) is the orthogonal representation group in 3-1and r denotes the second quantization. Murkov property
of euclidean
[4]. Let 0 be an open region in R4. For any positive or integrable
0 E CA E{aICAc}
Generalized
refZection property.
=
E{+%A}.
Let T = P(u(T))
and Z = r(M).
For every (YE
CR3
To = ZCY.
Moreover, T, E are unitary on L2(0, C, p) and TE = ZT, E2 = id. THEOREM 1. Let A(f) be a generalized Gaussian stochastic process indexed by S( R4)@‘, satisfying euclidean covariance, Markov property and generalized reflection property stated above. Then S2 defined by satisfies symmetry axiom, et&dean tion.
S2(.f@g) = EA(f)A(g) covariance and euclidean Hilbeti space structure condi-
Proof Symmetry and euclidean covariance are obvious. For the proof of Hilbert space structure condition we show that S2(h4U(T)f+
for f+ E S+(R4). If a = A(f+),
8
f+)
r
0
(111.3)
then (111.3) becomes E[(ZrZ)(Y]
> 0.
(111.4)
EUCLIDEAN FIELD THEORY WITHOUT REFLECTION Let n be the half-space x0 > 0, so dA = l? Q E CA Ti5 E Enc. Then we have [4]:
E[(Zz%)]
POSITIVITY
183
and AC is the half-space 2’ < 0. Since
= E[(c”z7Y)E{cx~C&}] = E[(ETZi)E{cr&3}] =
E[E{mG~C,3}E{4CR3}]
=
E[~TE{(YICE23}E{~ICR3}]
Since by generalized reflection property, we have that E[(z”zx)cr] Hence (111.3) holds. (111.3).
Euclidean
=
E[p{apR3}~*]
Hilbert space structure
>
0.
condition
easily follows from
Remark: The general result of [2] allows to reconstruct from the Schwinger function corresponding to euclidean field A(f), relativistic quantum theory on the indefinite inner product space with Hilbert structure. REFERENCES [l] [2] [3] [4] [5]
K. Ostenvalder, R. Schrader: Commun. Math. Phys. 31 (1973), 83; Commun. Math. Phys. 42 (1975), 281. L. Jakbbczyk, F. Strocchi: Commun. Math. Phys. 119 (1988), 529. G. Morchio, F. Strocchi: Ann. Inst. H. Poincari A33 (1980), 251. E. Nelson: “Probability theory and Euclidean field theory”, Lecl. Notes in Phys. 25, Springer, Berlin 1973. L. Jakbbczyk, F. Strocchi: /. Math. Phys. 29 (1988), 1231.
REPORTS
ON MATHEMATICAL
PHYSICS
No. 2
EUCLIDEAN FIELD THEORY WITHOUT REFLECTION POSITMTY. THE CASE OF PHOTON FIELD
Institute of Theoretical Physics, University of Wrodaw, Wroclaw, Poland (Received September 13, 1991 - Revised November 5, 1991) We study the properties of euclidean field theory corresponding to the free photon field in the covariant (non-positive) gauge.
1. Introduction
In QFT, positivity of the Wightman state W = {Wn} corresponds to the reflectionpositivity of the euclidean Schwinger state S = {Sn} [l]. As it was shown in [2], QFT without Wightman positivity [3] but satisfying so-called Hilbert space structure condition, still allows the euclidean continuation. The corresponding Schwinger state S satisfies euclidean covariance and symmetry, but instead of reflection positivity, S satisfies the following euclidean Hilbert space structure condition (for notation and ail details see [2]): IS(eFY 8 G+)l I PS(F+)PS(G+), where ps is a non-degenerate
Hilbert semi-norm on the Borchers algebra f3+ over S+ (R4) P.s(F+) I IF+ls.
The above condition allows also recovering of Wightman state satisfying Hilbert space structure condition. In this note, we apply the general strategy of [2] to the simple case of photon field in the covariant (non-positive) gauge. In this way we obtain Schwinger functions of the theory which are not reflection-positive. On the other hand they are Nelson-Symanzik positive, hence they are expectation values of some generalized stochastic process. In the paper we examine in details the properties of this process and generalize Nelson reconstruction theory [4] to the case of Euclidean photon field. It seems that the method of reconstruction developed in the paper can be applied only to the free field case. 2. Photon field in Gupta-Bleuler
gauge
The theory is defined by the two-point function (11.1)
180
L. JAKdBCZYK
with f,g E S(R4)04 and I+“(q)
= (-GP’“)@(q),
(11.2)
where F;(q) is the Fourier transform of two-point Wightman-function of massless scalar field. It is easy to see that IV2 is not Wightman positive, but satisfies Hilbert space structure condition IW2(f
8 c?)l 5 PkmPW(9)
(11.3)
with semi-norm pw defined by (11.4) pw is obviously S-continuous
and non-degenerate in_- the 1sense that kerpw = ker IV. Hence, by spectral property, pw is a distribution on S(R+“) and we can define 5(T) = PWW
for j(q) = j(q) t {so 2 0) E S(%+4). Applying the results of [2], we conclude that the corresponding Schwinger function Sz will satisfy euclidean Hilbert space structure condition with semi-norm p.#+) defined in terms of pw(f) by the construction P.df+)
(11.5)
= %1+>7
where f+(<)e- (4°~o-is”~)d
1+(q) = J so
Pdf+) =
(2 J &wG?~ljiri(4J),2)1’2. (11.6)
p=o
On the other hand, we can directly continue Wz to the euclidean points [5]. Let us denote by tS the points in the extended permuted tube Iex’J’ of the form zs = (iz’, T?) with z”, zlc E R (Schwingerpoinfs). Then the corresponding Euclidean points zE are given by XE
with I = diag(i, 1, l,l). in the following way
=
1-‘*S
In the case of scalar field, the Schwinger function Sz is defined S2(X:B,
X,“)
=
W2(IXf,
IX?>,
(11.7)
where Wz is the Wightman function analytic in I extJ’. Since Wz is invariant with respect to all complex Lorentz transformations, Sz is invariant with respect to the transformations L if the form L = I-GlI,
(11.8)
EUCLIDEAN FIELD THEORY WITHOUT REFLECIION POSITIVITY
181
where A E L+(C) is such that A maps Schwinger points onto the Schwinger points. One can check that A E L+(C) satisfying GA = AG has this property, and the corresponding L forms the group SO(4). In the vector case, the following definition W)
(11.9)
guarantees the covariance of S2 under SO(4) if W2 in L+(C)-covariant. case we obtain
Hence, in our
%W&W)
= c (W,,V, “IY”2
s2p,
,p2 ($7
$)
(W,,V,~“,“,V&
= @““2S(&
(11.10)
@)
with s(r) = J e- &0-iT.$)@(q)&_ Now we can discuss the problem of reflection-positivity be the time reflection. In the vector case T is represented (of),@)
= c
(11.11) of $. Let r(z”, Ti?)=(-z”, by the following operator
T’)
(11.12)
qW_fY(~~) ”
with M = diag(-1,
1, l,l).
Using u(r) we have S2(4f-)f+
Hence &(u(r)f+
@ 9+)
= @WC?,
(11.13)
@ 8+).
@ f+) is not positive definite. On the other hand IS2(4T)f+
(11.14)
@ cl+)1 5 ?%G+M9+)
with Pdf+)
= (5
J
d4~d4YS2@ - Yp+,,w+,(y))
1’2
p=o = (2 J p=o
&7m7+du~,‘)
1’2
so we again obtain (11.6). 3. Euclidean field theory and generalized
Nelson construction
Let (~2, C, p) be a probability space. If X is a topological vector space a generalized stochastic process over X is a stochastic process indexed by elements of X, i.e. for f E X A(f) is a random variable on (Q, C, CL). Let A be a generalized stochastic process over S(Rd)@k. If A is an open set in Rd, CA denotes a-algebra generated by A(f) with supp f C A. For an arbitrary subset of Rd we define cA = n A,>A
C,
182
L.JAKi)BCZYK
(where A’ is an open set containing A). If we denote by E{.jC4} conditional expectation with respect to CA, then the Murkovproperty of A means that (111.1)
E{QlCAC)] = E{Ql&4]
for any positive or integrable random variable cy in CA and any open set A c Rd [4]. Let us consider now the Schwinger function & discussed in Section II. By NelsonSymanzik positivity of Sz, (f, g) = &(f 18g) defines the Hilbert space inner product in SJ#Z4)@? Let ‘FIbe the real Hilbert space completion of SR(R4)@4 with respect to (, ). (‘Ii, (, )) defines the unique generalized Gaussian stochastic process A over S(&)@4 [4]. It means that we have probability space (Q, C, p) and the random variables A(f), f E S(&)@4 are Gaussian with covariance -IW.f)A(g)
= s A(f)( R
= s2c.f@ 9).
(111.2)
Using the properties of Sz we can obtain the following properties of euclidean field A: Euclidean covuriance. For every a E R4 and R E SO(4) there exists unitary operator T(a, R) defined on L2(0, C, CL)such that
T(a, R)A(f)T(a,
R)-’ = A&R)).
T(a, R) is given by T(z~(u, R)), w h ere u(u, R) is the orthogonal representation group in 3-1and r denotes the second quantization. Murkov property
of euclidean
[4]. Let 0 be an open region in R4. For any positive or integrable
0 E CA E{aICAc}
Generalized
refZection property.
=
E{+%A}.
Let T = P(u(T))
and Z = r(M).
For every (YE
CR3
To = ZCY.
Moreover, T, E are unitary on L2(0, C, p) and TE = ZT, E2 = id. THEOREM 1. Let A(f) be a generalized Gaussian stochastic process indexed by S( R4)@‘, satisfying euclidean covariance, Markov property and generalized reflection property stated above. Then S2 defined by satisfies symmetry axiom, et&dean tion.
S2(.f@g) = EA(f)A(g) covariance and euclidean Hilbeti space structure condi-
Proof Symmetry and euclidean covariance are obvious. For the proof of Hilbert space structure condition we show that S2(h4U(T)f+
for f+ E S+(R4). If a = A(f+),
8
f+)
r
0
(111.3)
then (111.3) becomes E[(ZrZ)(Y]
> 0.
(111.4)
EUCLIDEAN FIELD THEORY WITHOUT REFLECTION Let n be the half-space x0 > 0, so dA = l? Q E CA Ti5 E Enc. Then we have [4]:
E[(Zz%)]
POSITIVITY
183
and AC is the half-space 2’ < 0. Since
= E[(c”z7Y)E{cx~C&}] = E[(ETZi)E{cr&3}] =
E[E{mG~C,3}E{4CR3}]
=
E[~TE{(YICE23}E{~ICR3}]
Since by generalized reflection property, we have that E[(z”zx)cr] Hence (111.3) holds. (111.3).
Euclidean
=
E[p{apR3}~*]
Hilbert space structure
>
0.
condition
easily follows from
Remark: The general result of [2] allows to reconstruct from the Schwinger function corresponding to euclidean field A(f), relativistic quantum theory on the indefinite inner product space with Hilbert structure. REFERENCES [l] [2] [3] [4] [5]
K. Ostenvalder, R. Schrader: Commun. Math. Phys. 31 (1973), 83; Commun. Math. Phys. 42 (1975), 281. L. Jakbbczyk, F. Strocchi: Commun. Math. Phys. 119 (1988), 529. G. Morchio, F. Strocchi: Ann. Inst. H. Poincari A33 (1980), 251. E. Nelson: “Probability theory and Euclidean field theory”, Lecl. Notes in Phys. 25, Springer, Berlin 1973. L. Jakbbczyk, F. Strocchi: /. Math. Phys. 29 (1988), 1231.