We investigate relations between the group of symmetries of a subspace of the symmetric tensor product of a separable Hilbert space and a representation of the partition function of a quantum entangled family of particles associated with this subspace. Consider a system of
N noninteracting identical bosons. In general, the system is described by the
N-fold symmetric tensor product
E⊙N, where
E is the Hilbert space that describes the one-particle system. In some cases, the system can be described by some subspace of
E⊙N. We consider the case in which this subspace is the closure of the linear span of some subset of the eigenbasis of the Hamiltonian of the system. We describe the group of symmetries
S such that the partition function of the system is
S-symmetric.
In our previous work, we proposed a mathematical framework for -symmetric quantum theory, and in particular constructed a Krein space in which -symmetric operators would naturally act. In this work, we explore and discuss various general consequences and aspects of the theory defined in the Krein space, not only spectral properties and -symmetry breaking but also several issues, crucial for the theory to be physically acceptable, such as time evolution of state vectors, probability interpretation, uncertainty relation, classical–quantum correspondence, completeness, existence of a basis, and so on. In particular, we show that for a given real classical system we can always construct the corresponding -symmetric quantum system, which indicates that -symmetric theory in the Krein space is another quantization scheme rather than a generalization of the traditional Hermitian one in the Hilbert space. We propose a postulate for an operator to be a physical observable in this framework.
We provide a mathematical framework for -symmetric quantum theory, which is applicable irrespective of whether a system is defined on or a complex contour, whether symmetry is unbroken, and so on. The linear space in which -symmetric quantum theory is naturally defined is a Krein space constructed by introducing an indefinite metric into a Hilbert space composed of square integrable complex functions in a complex contour. We show that in this Krein space every -symmetric operator is -Hermitian if and only if it has transposition symmetry as well, from which the characteristic properties of the -symmetric Hamiltonians found in the literature follow. Some possible ways to construct physical theories are discussed within the restriction to the class K ( H ).