1. Introduction.In quantum field theory we must consider the Hilbert space having non-countable bases which corresponds to a sequence of non-negative integers (n, n.,...).Since we can construct one-to-one mapping from the set of the sequences (n, n,...) onto the points in 0, 1 interval [8, we can identify these bases to [0, 1 interval.Let 7 denote a point in 0, 1 interval and let r be the element of the Hilbert space which corresponds to y.The element of this Hilbert space is usually represented by the formulae flCrrdt( 7) and Cr, in [3], E4] and [6], where C, Cr are constants, and d/(') is a measure on E0, 1].By single dz(r), however, we cannot represent every element of this Hilbert space.That is to say, by a continuous measure dz('), we cannot represent the element of the second form.On the other hand by the second form, we cannot represent the element of the first form.
Introduction.In quantum field theory we must consider the Hilbert space having non-countable bases which corresponds to a sequence of non-negative integers (n, n.,...).Since we can construct one-to-one mapping from the set of the sequences (n, n,...) onto the points in 0, 1 interval [8, we can identify these bases to [0, 1 interval.Let 7 denote a point in 0, 1 interval and let r be the element of the Hilbert space which corresponds to y.The element of this Hilbert space is usually represented by the formulae flCrrdt( 7) and Cr, in [3], E4] and [6], where C, Cr are constants, and d/(') is a measure on E0, 1].By single dz(r), however, we cannot represent every element of this Hilbert space.That is to say, by a continuous measure dz('), we cannot represent the element of the second form.On the other hand by the second form, we cannot represent the element of the first form.
1. This paper, although mathematical in content, is motivated by quantumtheoretical considerations. The states of a quantum-mechanical system are usually described by vectors f of norm 1 in some Hilbert space A, and we assume explicitly that to every unit vector f corresponds a state of the system. This correspondence, however, is not one-to-one. In fact, the vectors which describe the same state form a ray f (in Weyl's terminology, cf. [13], p. 4 and p. 20),1 i.e. a set consisting of all vectors f = Tfo where fo is a fixed unit vector in & and r any complex number of modulus 1. (Every vector f in f will be called a representative of the ray f.) We have therefore a one-to-one correspondence between quantum states and rays, and every significant statement in Quantum Theory is a statement about rays. The transition probability from a state f to a state g equals (f, I)'2 where f, g are representatives of the rays f, g respectively. This suggests the introduction of the inner product of two rays by the definition
It is shown that merely the decision to represent the fundamental events of quantum theory by vectors in a Hilvert space uniquely determines the form of any probability measure over the set of events. The proof proceeds by showing that any nonnegative meausre on the vectors of a Hilbert space, for which the sum of the measures on the vectors of any complete orthonormal set is 1, takes the usual form damanded by quantum theory. This result represents a considerable strengthening of one of the consequences of Gleason’s theorem.
We demonstrated in the previous chapter (Chapter 1) that the evaluation of matrix elements between states represented by the Slater determinant can be evaluated unambiguously using Slater’s rule [1, 28, 40]. While the determinantal representation and the evaluation matrix elements between the states are quite straightforward, this procedure can eventually be ineffective and cumbersome with increasing size of the system. The occupation number formalism or second quantization technique introduced by Dirac in quantum field theory, in this context has been found to be quite formidable and elegant. This formalism not only allows us to represent the states and the operators in a compact and convenient form but also enables us to deal with systems containing variable numbers of particles without explicitly specifying the number of particles. It implicitly assumes the existence of an unspecified number of functions in a single-particle basis. These one-particle functions, which constitute the Hilbert space for N particles, are generally chosen to be spinorbital because of their orthogonality. In the case of fermions, the many-particle basis is the anti-symmetric product of N one-particle basis functions (Slater determinant). The vector space consisting of all possible anti-symmetric Hilbert spaces is referred to as the Fock space, in which the operators and functions are represented without specifying the number of particles.
Hilbert space and inner product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ( x ) which depends upon the position variable x , or by an alternative function ψ( p ) of the momentum variable p . It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars , which are complex numbers, are the elements of the quantum Hilbert space H . (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.) If α is any scalar (complex number), the ket corresponding to the wave function ɑψ( x ) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to ø( x )+ψ( x ) is denoted by |ø〉+|ψ〉 or |ψ〉+|ø〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ( x ) and ( p ), so that αø( x )+βψ( x ) and αϕ( p ) + βψ( p ) correspond to the same quantum state α|ψ〉+β|ø〉.
Abstract. In the present paper, we consider the marginal entropy of the a pure state in the tensor product of two Hilbert spaces. The expectation of the marginal entropy of a random pure state with respect to the Haar measure of unitary transformation on the tensor product Hilbert space depends on group integrals of unitary group. For group integrals of ordinary representation matrix elements of U(n), we provide a elementary method to reobtain Weingarton’s result of asymptotical behavior of group integrals. For group integrals of matrix elements of irreducible representations of U(n),we generalize the Weyl-Schur duality theorem and get an algorithm. 1. Mathematical Preliminaries We adopt the following standard conventions of quantum mechanics: The states of an n-level quantum system (and we include the case infinity) are represented by an n-dimensional complex Hilbert space H. In this familiar representation, the states of a composite of two systems A and B say, the first one A n-level, and the second B m-level, is then represented by the tensor product of the respective Hilbert spaces HA and HB. As a result, the composite system (AB) is an (nm)-level system. This also makes sense in the infinite case where we then use standard geometry for Hilbert space, and suitable choices of orthonormal bases (ONBs). For a fixed system with Hilbert space H, the corresponding pure quantum states are vectors in H of norm one, or rather equivalence classes of such vectors: Equivalent vectors v and v ′ in H yield the same rank-one projection operator P, i.e., the projection of H onto Key words and phrases. random pure state, marginal entropy,group integral, Weyl-Schur duality
In this series of papers we study whether the framework of a Hilbert space is adequate to the quantum theory of fields. Especially in the present paper we inquire what are the characteristic features of field theory, and study how these features can be described in this framework. For this purpose we examine the character of additive field quantities, which are used in this paper in place of field operators, and the relation between a system extending over the whole space and one restricted within a finite volume. After these considerations, we may conclude that there is a sufficient reason to treat the quantum theory of fields within the framework of a Hilbert space.