Fock Space (1)
The preceding chapters dealt with the non-commutative analogues of discrete r.v.’s, then of real valued r.v.’s, and we now begin to discuss stochastic processes. We start with the description of Fock space (symmetric and antisymmetric) as it is usually gi
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Fock Space (1) The preceding chapters dealt with the non-commutative analogues of discrete r.v.'s, then of real valued r.v.'s, and we now begin to discuss stochastic processes. We start with the description of Fock space (symmetric and antisymmetric) as it is usually given in physics books. Then we show that boson Fock space is isomorphic to the L 2 space of Wiener measure, and interpret on Wiener space the creation, annihilation and number operators. We proceed with the Poisson interpretation of Fock space, and the operator interpretation of the Poisson multiplication. We conclude with multiplication formulas, and the useful analogy with "toy Fock space" in chapter II, which leads to the antisymmetric (Clifford) multiplications. All these operations are special cases of Maassen's kernel calculus (§4). §1. BASIC DEFINITIONS Tensor product spaces 1 Let 1t be a complex Hilbert space. We consider its n-fold Hilhert space tensor power Jt®n , and define (1.1) (1.2)
Ut o ••• OUn
u1 " ... " Un
1 = 1 n.
1 = In.
L
Ua-(1)
® ... ® Uu(n),
o-EGn
L
eu Uq (1)
® ... ® Uu (n)
'
o-EGn
Gn denoting the group of permutations
Cl
of {1, ... , n}, with signature
eu.
The symbol
I\ is the exterior product, and o the symmetric product. The closed subspace of Jt®n
generated by all vectors (1.1) ( resp. (1.2)) is called the n-th symmetric (antisymmetric) power of 1t, and denoted by 1i 0 n (1!1\n). Usually, we denote it simply by 1-ln, adding o or I\ only in case of necessity. We sometimes borrow Wiener's prohabilistic terminology and call it the n-th chaos over 1t. We make the convention that 1-lo = C, and the element 1 E C is called the vacuum vector and denoted by 1 . Given some subspace U of 1t (possibly 1i itself) we also define the incomplete n-th chaos over U to be the subspace of 1tn consisting of linear combinations of products (1.1) or (1.2) with Ui EU. It does not seem necessary to have a general notation for it. In the physicists' use, if 1i is the Hilhert space describing the state of one single particle, 1tn descrihes a system of n particles of the same kind and is naturally called the n-particle space. The fact that these objects are identical is expressed in quantum P.-A. Meyer, Quantum Probability for Probabilists © Springer-Verlag Berlin Heidelberg 1993
56
IV. Fock space {1)
mechanics by symmetry properties of their joint wave function w.r.t. permutations, symmetry and antisymmetry being the simplest possibilities. Mathematics allow other "statistics", but they have not (yet ?) been observed in nature. We tend to favour bosons over fermions in these notes, since probabilistic interpretations are easier for bosons. One can find in the Iiterature two useful norms or scalar products on 1in , differing by a scalar factor. The first one is that induced by 1-(.®n,
< UI @ ••. @ Un, VI ® ... @ Vn > = < UI, VI > ... .
(1.3)
According to (1.2) we then have, in the antisymmetric case for instance (with u and r ranging over the permutation group Gn )
< UI "
••• "
Un' VI " ••• " Vn
2 > = (-\ n. )
L eo- e.,.
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