From Stochastic Differential Equations to Quantum Field Theory
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1
(N )
1 −2 ||f ||2
−1 2
S ′ (Rd )
;
(3)
where ||f ||2 −1 =
2
f (x)(1 − ∆)− 2 (x − y)f (y)dxdy.
0 0 For a various aspects of the process(es) ξt (resp. ξt (x0 )) we refer to [7, 24, 1, 2]. Example 2 Scalar fields as invariant measures of the Ito -type SDEs. Let us consider the following stochastic differential equations: 0 ǫ ǫ,0 dLǫ, t = (1 − ∆) Lt dt + (1 − ∆)
S ′ (Rd )
dµGN (η )ei(η,f ) = e− 2 ||f ||2 .
1
2
(6)
1 ], let us consider the following partial (pseudo)-differential stochastic For λ ∈ (0, 2 equation (1 − ∆)λ ϕλ = η. (7)
The solution of (7), given by the stochastic integral ϕλ = (1 − ∆)−λ ∗ η (8)
1 , ϕλ is identical to the free Nelson field. In particular, for λ = 2 3.B Now, let η be a S ′ (Rd ) ⊗ RN -valued Gaussian white noise and let τ be some real (orthogonal) representation of rotation group (S )O (d) in the space RN and let D be τ -covariant differential operator. Providing that D is such that corresponding Green function D −1 can be properly defined, it follows that the stochastic integral A ≡ D −1 ∗ η gives the solution of the following covariant stochastic differential equation ˜ A = η. D (10) ˜ is the adjoint of D in the canonical pairing S ′ < ·, · >S . In particular, taking where D d = 3, τ = D1 ⊕ D1 and
Abstract Covariant stochastic partial (pseudo)-differential equations are studied in any dimension. In particular a large class of covariant interacting local quantum fields obeying the Morchio-Strocchi system of axioms for indefinite quantum field theory is constructed by solving the analysed equations. The associated random cosurface models are discussed and some elementary properties of them are outlined.
ǫ−1 2
dBt
(4)
where dBt is the cylindric S ′ (Rd+1 )-valued Brownian motion and the (regularizing) parameter ǫ ∈ (0, 1]. By simple computations it follows that the invariant measure for (N ) (L) ,0 all of the equations (4) is equal to the free Nelson field µ0 . The law L(L1 t ) ≡ µ0 of the stationary solution of (4), called the free Langevin field, is easily seen as centered Gaussian probability measure on (S ′ (Rd+1 ), B(S ′ (Rd+1 ))) characterized by
0 0 has a stationary solution ξt which has a law L(ξt ) = µN 0 easily identified with the free N Nelson field, i.e. µ0 is a Gaussian probability measure on (S ′ (Rd+1 ), B(S ′ (Rd+1 )) ≡ cylinder σ -algebra) (≡ Gaussian generalized random field) with the mean equal to zero and covariance 0 0 E ξ0 (x)ξt (y )
Example 3 Free quantum fields as stochastic integrals. 3.A Let η be a Gaussian white noise on the space S ′ (Rd ), i.e. η is distributed according to the Gaussian measure dµGN characterized by:
S ′ (Rd+1 )
dµ0 (ϕ)ei(ϕ,f ) = e− 2 ||f ||−1,−2 ;
(L)
1
2
(5)
where ||f ||2 −1,−2 =
2 f (s, x)(−∂0 + (1 − ∆)2 )−1 (s − t, x − y )f (t, y )dsdxdtdy. (L)
The corresponding to dµ0 generalized random field is (sharp)-Markov in the computer time direction t and (germ)-Markov in the other directions. The study of the equations 4 is a part of the so called stochastic quantisation programme [20, 17, 15, 23, 27, 28]. 2
1 strict connection of the Euclidean (bosonic) quantum field theory with (infinitedimensional) Stochastic Analysis objects and concepts [21, 22] is well known in different situations. Let us recall few of them. Example 1 Scalar fields as solutions of the Ito -type SDEs Let Bt be a cylindric version of S ′ (Rd )-valued Brownian motion (where S ′ (Rd ) stands for the space of real tempered distributions) i.e. for any f ∈ S (Rd ) the coordinate
m 0 0 0 b∂z −b∂y 0 m 0 −b∂z 0 b∂x 0 0 m b∂y −b∂x 0 0 c∂z −c∂y m 0 0 −c∂z 0 c∂x 0 m 0 c∂y −c∂x 0 0 0 m
(11)
with b2 = c2 = 1, and bc = −1 it follows that the stochastic integral D −1 ∗ η for η being pure Gaussian white noise gives two independent copies of two real massive(with the mass m) Euclidean Proca fields. For a systematic approach to such constructions see [12] and for a particular application to the free EM4 fields [8, 4, 5]. The particular features of the above listed examples are: the linearity of the corresponding equations and Gaussianity of the corresponding noise. These features lead to the Gaussian solutions, therefore not very interesting from the point of view of physics. The interesting physics seems to be described by non-Gaussian examples. 3
(N )
1 −2 ||f ||2
−1 2
S ′ (Rd )
;
(3)
where ||f ||2 −1 =
2
f (x)(1 − ∆)− 2 (x − y)f (y)dxdy.
0 0 For a various aspects of the process(es) ξt (resp. ξt (x0 )) we refer to [7, 24, 1, 2]. Example 2 Scalar fields as invariant measures of the Ito -type SDEs. Let us consider the following stochastic differential equations: 0 ǫ ǫ,0 dLǫ, t = (1 − ∆) Lt dt + (1 − ∆)
S ′ (Rd )
dµGN (η )ei(η,f ) = e− 2 ||f ||2 .
1
2
(6)
1 ], let us consider the following partial (pseudo)-differential stochastic For λ ∈ (0, 2 equation (1 − ∆)λ ϕλ = η. (7)
The solution of (7), given by the stochastic integral ϕλ = (1 − ∆)−λ ∗ η (8)
1 , ϕλ is identical to the free Nelson field. In particular, for λ = 2 3.B Now, let η be a S ′ (Rd ) ⊗ RN -valued Gaussian white noise and let τ be some real (orthogonal) representation of rotation group (S )O (d) in the space RN and let D be τ -covariant differential operator. Providing that D is such that corresponding Green function D −1 can be properly defined, it follows that the stochastic integral A ≡ D −1 ∗ η gives the solution of the following covariant stochastic differential equation ˜ A = η. D (10) ˜ is the adjoint of D in the canonical pairing S ′ < ·, · >S . In particular, taking where D d = 3, τ = D1 ⊕ D1 and
Abstract Covariant stochastic partial (pseudo)-differential equations are studied in any dimension. In particular a large class of covariant interacting local quantum fields obeying the Morchio-Strocchi system of axioms for indefinite quantum field theory is constructed by solving the analysed equations. The associated random cosurface models are discussed and some elementary properties of them are outlined.
ǫ−1 2
dBt
(4)
where dBt is the cylindric S ′ (Rd+1 )-valued Brownian motion and the (regularizing) parameter ǫ ∈ (0, 1]. By simple computations it follows that the invariant measure for (N ) (L) ,0 all of the equations (4) is equal to the free Nelson field µ0 . The law L(L1 t ) ≡ µ0 of the stationary solution of (4), called the free Langevin field, is easily seen as centered Gaussian probability measure on (S ′ (Rd+1 ), B(S ′ (Rd+1 ))) characterized by
0 0 has a stationary solution ξt which has a law L(ξt ) = µN 0 easily identified with the free N Nelson field, i.e. µ0 is a Gaussian probability measure on (S ′ (Rd+1 ), B(S ′ (Rd+1 )) ≡ cylinder σ -algebra) (≡ Gaussian generalized random field) with the mean equal to zero and covariance 0 0 E ξ0 (x)ξt (y )
Example 3 Free quantum fields as stochastic integrals. 3.A Let η be a Gaussian white noise on the space S ′ (Rd ), i.e. η is distributed according to the Gaussian measure dµGN characterized by:
S ′ (Rd+1 )
dµ0 (ϕ)ei(ϕ,f ) = e− 2 ||f ||−1,−2 ;
(L)
1
2
(5)
where ||f ||2 −1,−2 =
2 f (s, x)(−∂0 + (1 − ∆)2 )−1 (s − t, x − y )f (t, y )dsdxdtdy. (L)
The corresponding to dµ0 generalized random field is (sharp)-Markov in the computer time direction t and (germ)-Markov in the other directions. The study of the equations 4 is a part of the so called stochastic quantisation programme [20, 17, 15, 23, 27, 28]. 2
1 strict connection of the Euclidean (bosonic) quantum field theory with (infinitedimensional) Stochastic Analysis objects and concepts [21, 22] is well known in different situations. Let us recall few of them. Example 1 Scalar fields as solutions of the Ito -type SDEs Let Bt be a cylindric version of S ′ (Rd )-valued Brownian motion (where S ′ (Rd ) stands for the space of real tempered distributions) i.e. for any f ∈ S (Rd ) the coordinate
m 0 0 0 b∂z −b∂y 0 m 0 −b∂z 0 b∂x 0 0 m b∂y −b∂x 0 0 c∂z −c∂y m 0 0 −c∂z 0 c∂x 0 m 0 c∂y −c∂x 0 0 0 m
(11)
with b2 = c2 = 1, and bc = −1 it follows that the stochastic integral D −1 ∗ η for η being pure Gaussian white noise gives two independent copies of two real massive(with the mass m) Euclidean Proca fields. For a systematic approach to such constructions see [12] and for a particular application to the free EM4 fields [8, 4, 5]. The particular features of the above listed examples are: the linearity of the corresponding equations and Gaussianity of the corresponding noise. These features lead to the Gaussian solutions, therefore not very interesting from the point of view of physics. The interesting physics seems to be described by non-Gaussian examples. 3
