On Various Functional Representations of the Space of Schwartz Operators
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ON VARIOUS FUNCTIONAL REPRESENTATIONS OF THE SPACE OF SCHWARTZ OPERATORS G. G. Amosov
UDC 517.98, 530.245
Abstract. In this paper, we discuss various representations in which the space S of Schwartz operators turns into the space of test functions, whereas the dual space S turns into the space of generalized functions. Keywords and phrases: quantum tomography, functional representation, space of Schwartz operators. AMS Subject Classification: 81P16, 46F99
1. Introduction. As is known, the ideal of nuclear operators S1 (H) is a predual space for the algebra of all bounded operators B(H) in the Hilbert space H. If we identify B(H) with the algebra of quantum observables, then the duality relation allows one to consider observables A ∈ B(H) as functionals on the set of quantum states SH consisting of positive operators ρ ∈ S1 (H) normalized by the condition Tr(ρ) = 1; the action of a functional is defined by the formula A, ρ = Tr(ρA);
(1)
here A ∈ B(H) and ρ ∈ S(H). This approach is sufficient for finite-dimensional spaces, dim H < +∞. In infinite-dimensional Hilbert spaces, all self-adjoint operators A (even unbounded) must be considered as observables. In order to save the dual structure between states and observables, it is necessary to distinguish a certain linear subspace S ⊂ S1 (H), and endow it with a topology in such a way that the dual space S will contain unbounded self-adjoint operators at least of a certain class, and the duality relation must turn into (1) for A ∈ B(H). In the recent work [9], the notion of the space of Schwartz operators S was introduced. It was shown that S has the structure of a Frechet space, and the dual space S contains, in particular, unbounded operators, which are polynomials of the coordinate and momentum operators. Our task is to study representations of the space of Schwartz operators S in various functional spaces. We consider such functional spaces as spaces of test functions, so that the dual representation of S will consist of generalized functions over them. The construction of functional representations of operators is directly related to the tomographic representation of quantum mechanics. Under the tomographic representation, we mean a certain procedure, which allows one to pass from operators of states and observables to functions of some number of variables. The development of the tomographic representation led to the following formalism ˆ (x) and D(x) ˆ (see [5, 10]). In a Hilbert space H, we introduce sets of operators U depending on an ˆ n-dimensional vector x = (x1 , x2 , . . . , xn ) so that to the operator A on H, the function ˆ (x)Aˆ (2) fAˆ (x) = Tr U ˆ so that the operator can be is assigned. This function is called the symbol of the the operator A, restored by its symbol by the formula ˆ (3) Aˆ = D(x)f ˆ (x)dx. A ˆ (x) and D(x) ˆ The families U are called the dequantizer and the quantizer, respectively. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 151, Quantum Probability
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