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ESk~l) -I. Let y > O Then the following defines a unique quasi-free state o f the C C R or C A R for one degree of freedom at each time: 8 = n(k) (subject to the single normalization B = fo 1o(k) 12e-ikt (e-k) 2+y2 condition lip (k)12[ (w-k) 2+y2]-Idk = i) and J dk (-+n(k) +l) p2 (k) [ (m-k) 2+~23- le-iktdk. O This class of models is less singular than the model of Ford, Kac and M a z u r which P(t) does not exist except as a distribution. via the Wight_man reconstruction dA(t) theor~n, = -iwA(t) The operators A(t), obey the damped - yA(t) stochastic [9], in constructed equation + a(t) where a(t) is the noise operator a(t) = ~ k e-ikta(k)p(k) the ~(k) being independent oscillators b a t h is d e s c r i b e d b y p(k).

In Local Quantum Theor~ (R. ), Academic Press, 1969. M. L. Hudson: Multivariate Anal. ~, 107-123 Quantum Mechanical Wiener Processes. (1978). [4] T. Hida: Brownian Motion, Springer-Verlag, [5] C. F. F. Wilde. Funct. Anal. F. Wilde. 12-21 [7] [8] The Ito-Clifford Integral. The Free Fermion field as a Markov field. Markov Property. F. Streater. Commun. Maths. Phys. J. Funcl. Anal. 15, The Ito-Clifford Integral III (to appear). Damped Oscillator with Quantum Noise. J. Math. Phys. 6__, 504-515 Statistical Mechanics of coupled oscillators.

A(f) (lower sign) are unbounded, the set of o p e r a t o r s {a (f) : f E K} g e n e r a t e This is the CAR algebra. The B o s o n operators but subject to w e l l - s t u d i e d c o n d i t i o n s can also b e used to define a unique c*-algebra, the C C R a l g e b r a [11. O f the m a n y r e p r e s e n t a t i o n s of the CCR or the CAR, the m o s t i m p o r t a n t ones are the Fock representations. e. a(f)~ F = O for all f £ K. Note that the o n e - p a r t i c l e space K is not the same as the space ~ o p e r a t o r s act.

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