It is well known that the canonical commutation relation [x, p] = i can be realized only on an infinite-dimensional Hilbert space. While any finite set of experimental data can also be explained in terms of a finite-dimensional Hilbert space by approximating the commutation relation, Occam's razor prefers the infinite-dimensional model in which [x, p] = i holds on the nose. This reasoning one will necessarily have to make in any approach which tries to detect the infinite-dimensionality. One drawback of using the canonical commutation relation for this purpose is that it has unclear operational meaning. Here, we identify an operationally well-defined context from which an analogous conclusion can be drawn: if two unitary transformations U, V on a quantum system satisfy the relation V−1U2V = U3, then finite-dimensionality entails the relation UV−1UV = V−1UVU; this implication strongly fails in some infinite-dimensional realizations. This is a result from combinatorial group theory for which we give a new proof. This proof adapts to the consideration of cases where the assumed relation V−1U2V = U3 holds only up to ε and then yields a lower bound on the dimension.
It is well known that the canonical commutation relation [x, p] = i can be realized only on an infinite-dimensional Hilbert space. While any finite set of experimental data can also be explained in terms of a finite-dimensional Hilbert space by approximating the commutation relation, Occam's razor prefers the infinite-dimensional model in which [x, p] = i holds on the nose. This reasoning one will necessarily have to make in any approach which tries to detect the infinite-dimensionality. One drawback of using the canonical commutation relation for this purpose is that it has unclear operational meaning. Here, we identify an operationally well-defined context from which an analogous conclusion can be drawn: if two unitary transformations U, V on a quantum system satisfy the relation V−1U2V = U3, then finite-dimensionality entails the relation UV−1UV = V−1UVU; this implication strongly fails in some infinite-dimensional realizations. This is a result from combinatorial group theory for which we give a new proof. This proof adapts to the consideration of cases where the assumed relation V−1U2V = U3 holds only up to ε and then yields a lower bound on the dimension.
This paper contributes to a recent series discussing quantum mechanics defined on a finite-dimensional Hilbert space in which Weyl's commutation relation for unitary operators holds. In an earlier paper, Santhanam and Tekumalla (1976) showed that the commutation relation for hermitian operators with a bounded spectrum tends to Hdsenberg's standard canonical commutation relation as the spectrum becomes continuous and the dimension n -> 00. The present paper offers a formulation which is coordinate free in the limit n -> 00 and makes the limiting procedure especially ransparent.
Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.
The claim that unitary transformations in quantum mechanics correspond to the canonical transformations of classical mechanics is not correct. The spectra of operators produced by unitary transformation of Cartesian coordinate and momentum operators (q, k) are necessarily continuous over the entire real domain of their eigenvalues. Operators with spectra which are not everywhere continuous are generated from (q, k) by one-sided unitary transformations U for which U†U = 1 but for which UU† commutes with either q or k (but not both). If UU† commutes with k, the new coordinates and momenta (r, s) satisfy commutation relations [sm, rn] = 2πi1δm,n, [sm, sn] = 0, but [rm, rn] ≠ 0; (r, s) are canonical only for one-dimensional systems. The properties of one-sided unitary transformations are described; they are characterized by φ(K), the eigenvalue of UU†. The one-dimensional case for which the one-sided unitary transformation is canonical is discussed in detail. A prescription is given for obtaining the operator canonically conjugate to any one-dimensional observable. Generalization to higher dimensions is also discussed.
In Chapter 4 we saw that the canonical commutation relations force us to use a space of states of infinite dimension, in which rigor would require the use of advanced mathematical tools. Fortunately, physicists generally need only to carry the results for finite dimension over to infinite dimension with some simple modifications which we shall indicate here, without embarking on sophisticated mathematics. Nevertheless, it is useful to be aware of the lapses in rigor which are customarily made in physics in order to avoid possible unpleasant surprises. The objective of this chapter is, on the one hand, to present some concrete examples illustrating the new features which arise in infinite dimension and, on the other, to give the rules for practical calculations, in particular to write down the spectral decomposition of Hermitian and unitary operators. The mathematics we use is a bit more detailed than commonly found in most quantum mechanics textbooks. The reader interested purely in the practical aspects can proceed directly to Section 7.3, where the results essential for later on are summarized. Hilbert spaces Definitions The space of states of quantum mechanics is a Hilbert space ℌ, which in general is of infinite dimension. The axiomatic definition of a Hilbert space is the following. It is a vector space which, for the needs of quantum mechanics, is defined on complex numbers. The vectors of this space are denoted |ϕ〉. […]
In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let H be a Hilbert space and let B ( H ) be the set of all bounded operators on H . Here a notable point is that linear operators on finite - dimensional Hilbert spaces are automatically bounded, whereas in general they are not. Thus we impose boundedness as an extra requirement, beyond linearity. This is very convenient, because as in the finite-dimensional case, B ( H ) is a C*-algebra, cf. §C.1. At the same time, assuming boundedness involves no loss of generality whatsoever, since we can alway replace closed unbounded operators by bounded ones through the bounded transform , as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A fundamental problem which arises in determining whether two quantum mechanical systems are essentially identical is whether a unitary or antiunitary transformation exists which maps one set of dynamical variables into another. Since an elementary dynamical system is specified by giving an irreducible set of dynamical variables, we are led to investigate the following problem: Given two irreducible sets of operators with a one-to-one correspondence between them, find the algebraic properties of the two sets which make it possible to infer the existence of a unitary or antiunitary operator relating them. A series of theorems is obtained from such considerations for finite dimensional spaces. It is shown that if the second set of operators contains some of the algebraic properties of the first set, the two sets are related by a similarity transformation. By altering the requirements, this transformation is a unitary transformation. Indications are also given to show how the theorems can be extended to Hilbert spaces. The rigorous statements of the theorems and the proofs will be given in a second paper. Finally, in the Appendix there is given a definition of invariance of elementary quantum-mechanical systems based on the above theorems, giving the same results as Wigner's definition in terms of transition probabilities.