The definition of quantum states in terms of tagged vector spaces is generalized to incorporate the spatiotemporal and spin degrees of freedom. Considering a tagged vector space where the index space is a function space, representing the additional degrees of freedom, we obtained axioms for the tags that include a completeness condition expressed in terms of a functional integral with an abstract functional integration measure. Using these axioms, we derive a generating functional for the moments of this functional integration measure. These moments are then used to evaluate the functional integrals of Gaussian functionals, leading to expressions in accordance with those obtained as generalizations of equivalent integrals over a finite number of integration variables. For a Gaussian functional used as a probability distributions, we show that its moments, obtained with this functional integration measure, satisfy Carleman's condition, indicating that the measure is unique.
The definition of quantum states in terms of tagged vector spaces is generalized to incorporate the spatiotemporal and spin degrees of freedom. Considering a tagged vector space where the index space is a function space, representing the additional degrees of freedom, we obtained axioms for the tags that include a completeness condition expressed in terms of a functional integral with an abstract functional integration measure. Using these axioms, we derive a generating functional for the moments of this functional integration measure. These moments are then used to evaluate the functional integrals of Gaussian functionals, leading to expressions in accordance with those obtained as generalizations of equivalent integrals over a finite number of integration variables. For a Gaussian functional used as a probability distributions, we show that its moments, obtained with this functional integration measure, satisfy Carleman's condition, indicating that the measure is unique.
A generalisation is provided for the notion of tags, often found in physics notation. The properties of tags and their extractors are discussed. It leads to the definition of tagged vector spaces. It is applied in the context of quantum optics, to provide a mathematical description for Dirac notation as used in physics. The commutation relations for ladder operators and a symplectic phase space are derived as consequences of the properties of tagged vector spaces. To incorporate the spatiotemporal and spin degrees of freedom, tags are indexed by functions, leading to a functional representation.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
Using the quadrature bases that incorporate the spatiotemporal degrees of freedom, we develop a Wigner functional theory for quantum optics, as an extension of the Moyal formalism. Since the spatiotemporal quadrature bases span the complete Hilbert space of all quantum optical states, it does not require factorization as a tensor product of discrete Hilbert spaces. The Wigner functions associated with such a space become functionals and operations are expressed by functional integrals -- the functional version of the star product. The resulting formalism enables tractable calculations for scenarios where both spatiotemporal degrees of freedom and particle-number degrees of freedom are relevant. To demonstrate the approach, we compute examples of Wigner functionals for a few well-known states and operators.
All the chapters of the book have examined and studied various aspects of path integrals and Hamiltonians, which in turn exemplified different aspects of quantum mathematics. The principles of quantum mathematics, stated in Chapter 2, can be summarized as follows: • The fundamental degrees of freedom F form the bedrock of the quantum system. • A linear vector state space V based on the degrees of freedom F provides an exhaustive description of the quantum system. • Operators O, which includes the Hamiltonian H , represent the physical properties of the degree of freedom and act on the state space V . Observable quantities are the matrix elements of the physical operators. • A spacetime description of quantum indeterminacy is encoded in the Lagrangian L and the Dirac–Feynman formula relates it to the Hamiltonian. • The path integral provides a representation of all the physical properties of a quantum system. In particular, the path integral yields the correlation functions of the degrees of freedom as well as the probability amplitudes for quantum transitions. • The interconnection of the path integral with the underlying Hamiltonian and state space is a specific feature of quantum mathematics that distinguishes path integration from functional integration in general.
Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup E2(H) of bounded operators in Γ(H). This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries:
(i) every gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(Λ)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, Λ being a positive operator in H, ar,1≤r≤n are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C and [αrs] is a symmetric matrix in Mn(C);
(ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its E2(H)-parameters;
(iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure n-mode gaussian states which are completely entangled;
(iv) tomography of an unknown gaussian state in Γ(Cn) by the estimation of its E2(Cn)-parameters using O(n2) measurements with a finite number of outcomes.
Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.