Considering physics as it is done today one notices that physics separates into two formally and conceptually different parts. There is on the one hand classical physics that deals with macroscopic phenomena and there is on the other hand quantum physics that deals with microscopic phenomena. Penrose [96] described this situation as a disturbing analogy to ancient Greece, where two different sets of laws for earth and heaven were applied. So immediately the question arises if one really has to use different formalisms on different scales or if it is possible to describe physics in a unified way. In order to investigate this question one first has to consider how classical and quantum physics are related. Quantum mechanics results from classical physics by a procedure called quantization and classical physics is reobtained by taking the classical limit. Both procedures are heavily plagued by problems [61, 100]. In the first chapter the mathematical problems of quantization will be addressed and it will be described how the star product formalism circumvents these problems. The star product formalism has the advantage that there is no formal break if one goes from classical physics to quantum physics. This formal advantage and the resulting beauty is then used as a guiding principle for the further development. The first question that arises is if the spin can be described in the star product formalism. That this is indeed possible is shown in the second chapter. As a starting point the spin description with pseudoclassical mechanics as it was developed by Berezin is used. One can then construct a fermionic star product and apply it for deformation quantization of pseudoclassical mechanics. In analogy to the bosonic star product formalism one obtains spin Wigner functions that act as spin projectors. Besides the nonrelativistic case it is also possible to formulate Dirac theory with star products. The Clifford algebra of the gamma matrices is hereby described as a deformed version of a four dimensional Grassmann algebra. The fermionic star product in combination with the bosonic Moyal product leads to a supersymmetric star product formalism that can be used to describe supersymmetric quantum mechanics and in the relativistic case to describe the supersymmetric structure of Dirac theory. The other direction of generalization of the star product formalism is the application of star products in quantum field theory that is described in chapter three. After constructing a suitable normal product it is shown that the algebraic structures of perturbative quantum field theory appear also in the star product formalism, which is an expression of the algebra morphism of the operator and the star product formalism. But moreover the quantum group structure that was recently found in perturbative quantum field theory is shown to be a natural algebraic structure of the star product. The essential advantage of deformation quantization is that the classical limit has a well defined meaning. In the context of the spin description with star products this leads to the question of the classical limit of spin, or equivalently to the question of the physical status of pseudoclassical mechanics. In chapter four it is shown that this question is solved if one realizes that the fermionic sector together with the fermionic star product describe the underlying geometric structure. The deformation of a Grassmann algebra leads to a Clifford calculus that is equivalent to geometric algebra. The formulation of geometric algebra in the star product formalism is given in chapter four. One sees there that geometric algebra as the most fundamental geometric formalism that unifies all geometric structures that appear in physics can be described in a supersymmetric manner that parallels the bosonic star product structures. Having obtained a formulation of geometry with fermionic star products this formalism is then applied in the fifth chapter to physical problems. As examples for the application of geometric algebra in classical physics the rigid body and the Kepler problem are considered. In both cases the formalism of geometric algebra gives the most elegant formulation of the problem. In the quantum case one can then combine the fermionic star product formalism that describes the underlying geometric structure with the bosonic star product that describes the noncommuative structure of quantum mechanics. The result is a noncommutative version of geometric algebra that leads to a natural appearance of spin terms. The same idea applied on the phase space leads to the split in supersymmetric partner systems. Geometric algebra gives in this way a natural geometric foundation of supersymmetric quantum mechanics. Similarly one can interpret the hidden BRST-structure of classical mechanics that was found by E. Gozzi and M. Reuter in the path integral formalism from a star product point of view.
Considering physics as it is done today one notices that physics separates into two formally and conceptually different parts. There is on the one hand classical physics that deals with macroscopic phenomena and there is on the other hand quantum physics that deals with microscopic phenomena. Penrose [96] described this situation as a disturbing analogy to ancient Greece, where two different sets of laws for earth and heaven were applied. So immediately the question arises if one really has to use different formalisms on different scales or if it is possible to describe physics in a unified way. In order to investigate this question one first has to consider how classical and quantum physics are related. Quantum mechanics results from classical physics by a procedure called quantization and classical physics is reobtained by taking the classical limit. Both procedures are heavily plagued by problems [61, 100]. In the first chapter the mathematical problems of quantization will be addressed and it will be described how the star product formalism circumvents these problems. The star product formalism has the advantage that there is no formal break if one goes from classical physics to quantum physics. This formal advantage and the resulting beauty is then used as a guiding principle for the further development. The first question that arises is if the spin can be described in the star product formalism. That this is indeed possible is shown in the second chapter. As a starting point the spin description with pseudoclassical mechanics as it was developed by Berezin is used. One can then construct a fermionic star product and apply it for deformation quantization of pseudoclassical mechanics. In analogy to the bosonic star product formalism one obtains spin Wigner functions that act as spin projectors. Besides the nonrelativistic case it is also possible to formulate Dirac theory with star products. The Clifford algebra of the gamma matrices is hereby described as a deformed version of a four dimensional Grassmann algebra. The fermionic star product in combination with the bosonic Moyal product leads to a supersymmetric star product formalism that can be used to describe supersymmetric quantum mechanics and in the relativistic case to describe the supersymmetric structure of Dirac theory. The other direction of generalization of the star product formalism is the application of star products in quantum field theory that is described in chapter three. After constructing a suitable normal product it is shown that the algebraic structures of perturbative quantum field theory appear also in the star product formalism, which is an expression of the algebra morphism of the operator and the star product formalism. But moreover the quantum group structure that was recently found in perturbative quantum field theory is shown to be a natural algebraic structure of the star product. The essential advantage of deformation quantization is that the classical limit has a well defined meaning. In the context of the spin description with star products this leads to the question of the classical limit of spin, or equivalently to the question of the physical status of pseudoclassical mechanics. In chapter four it is shown that this question is solved if one realizes that the fermionic sector together with the fermionic star product describe the underlying geometric structure. The deformation of a Grassmann algebra leads to a Clifford calculus that is equivalent to geometric algebra. The formulation of geometric algebra in the star product formalism is given in chapter four. One sees there that geometric algebra as the most fundamental geometric formalism that unifies all geometric structures that appear in physics can be described in a supersymmetric manner that parallels the bosonic star product structures. Having obtained a formulation of geometry with fermionic star products this formalism is then applied in the fifth chapter to physical problems. As examples for the application of geometric algebra in classical physics the rigid body and the Kepler problem are considered. In both cases the formalism of geometric algebra gives the most elegant formulation of the problem. In the quantum case one can then combine the fermionic star product formalism that describes the underlying geometric structure with the bosonic star product that describes the noncommuative structure of quantum mechanics. The result is a noncommutative version of geometric algebra that leads to a natural appearance of spin terms. The same idea applied on the phase space leads to the split in supersymmetric partner systems. Geometric algebra gives in this way a natural geometric foundation of supersymmetric quantum mechanics. Similarly one can interpret the hidden BRST-structure of classical mechanics that was found by E. Gozzi and M. Reuter in the path integral formalism from a star product point of view.
The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative ⋆-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The considered class of deformations and the corresponding ⋆-products contains as a special cases all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of ⋆-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the ⋆-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. The equivalence of two formulations of quantum mechanics is proved by observing that the Wigner-Moyal transform has all properties of the tensor product. This observation allows writing many previous results found in the literature in a transparent way, from which the equivalence of the two formulations of quantum mechanics follows naturally.
In this chapter we study the application of geometric algebra to both nonrelativistic and relativistic quantum mechanics. We concentrate on the quantum theory of spin-1/2 particles, whose dynamics is described by the Pauli and Dirac equations. For interactions where spin and relativity are not important the dynamics reduces to that of the Schrödinger equation. There are many good textbooks describing this topic and we will make no attempt to cover it here. We assume, furthermore, that most readers have a basic understanding of quantum mechanics, and are familiar with the concepts of states and operators. Both the Pauli and Dirac matrices arise naturally as representations of the geometric algebras of space and spacetime. It is no surprise, then, that much of quantum theory finds a natural expression within geometric algebra. To achieve this, however, one must reconsider the standard interpretation of the quantum spin operators. Like much discussion of the interpretation of quantum theory, certain issues raised here are controversial. There is no question about the validity of our algebraic approach, however, and little doubt about its advantages. Whether the algebraic simplifications obtained here are indicative of a deeper structure embedded in quantum mechanics is an open question. In this chapter we only consider the quantum theory of single particles in background fields. Multiparticle systems are considered in the following chapter.
This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i.e. in phase space. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Wigner functions density -- matrices in a special Weyl representation -- and star products are the cornerstones of the formalism. The resulting framework is a rich source of physical intuition. It has been used to describe transport in quantum optics, structure and dynamics in nuclear physics, chaos, and decoherence in quantum computing. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative way to formulate and understand quantum mechanics, independent of the conventional Hilbert space or path integral approaches to the subject. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space variables. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory. The observables in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, only composed together in novel algebraic ways using star products. This treatise provides an introductory overview and supplementary material suitable for an advanced undergraduate or a beginning graduate course in quantum mechanics. Readership: Advanced undergraduates and graduates studying quantum mechanics.
We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.
This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory, including recent results. It is based on the algebraic approach in which the starting point is a star-algebra and on the geometric approach in which the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (or even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of an inclusive scattering matrix, which is closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied in the geometric approach as well. In particular, this result can be used to provide a simple derivation of the diagram technique for generalized Green functions. The notion of an inclusive scattering matrix makes sense in the geometric approach, although it seems that a definition of the conventional scattering matrix cannot be provided in this situation. The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of the observables.
Introduction The main theme of this survey is the quantization of classical mechanical systems in terms of algebraic geometry. The aim is thus to relate questions of mathematics and theoretical physics. We first recall briefly the main problems and methods which turn us to study the new subject. Quantization itself is the main problem of theoretical physics. The need to introduce and develop it was dictated by the creators of quantum theory. According to the ‘Copenhagen philosophy’, the physical predictions of a quantum theory must be formulated in terms of classical concepts (the first sentence of Woodhouse; here we quote the beginning of this survey). Thus in addition to the usual structures (Hilbert space, unitary transformations, selfadjoint operators …, see, for example, Landau and Lifschitz) any reasonable quantum theory has to admit an appropriate passage to a classical limit under which quantum observables are transferred to their classical analogues. However, as Dirac pointed out at the beginning of the quantum age, the correspondence between quantum theory and classical theory should be based not just on numerical coincidences taking place in the limit h → ∞, but on an analogy between their mathematical structures. Classical theory does approximate the quantum theory, but it does more – it supplies a framework for some interpretation of quantum theory. Using this idea, we can in general understand a quantization procedure as a correspondence between classical and quantum theories. In this sense, quantization of the classical mechanical systems is a movement in one direction, while taking the quasiclassical limit goes in the opposite direction.