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The Matrix Model for M Theory as an Exemplar of Trace (or Generalized Quantum) Dynamics

Stephen L. Adler

Published Mar 6, 1997• 3 citations
OpenAlex

Abstract

We show that the recently proposed matrix model for M theory obeys the cyclic trace assumptions underlying generalized quantum or trace dynamics. This permits a verification of supersymmetry as an operator calculation, and a calculation of the supercharge density algebra by using the generalized Poisson bracket, in a basis-independent manner that makes no reference to individual matrix elements. Implications for quantization of the model are discussed. Our results are a special case of a general result presented elsewhere, that all rigid supersymmetry theories can be extended to give supersymmetric trace dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a noncommutative trace class graded operator algebra.

Related papers

Trace dynamics models with global supersymmetry

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In Section 2.4, we illustrated the trace dynamics formalism by constructing the trace dynamics analog of a simple field theory model, in which a Dirac fermion interacts with a scalar Klein–Gordon field. Much of the recent literature in quantum field theory has concerned itself with supersymmetric theories, in which invariance under the Poincaré group has been extended to invariance under the graded Poincaré group, and theories of this type are considered likely to play a central role in the ultimate unification of the forces. Our aim in this chapter (which can be omitted on a first reading) is to show that the trace dynamics formalism naturally extends to globally supersymmetric theories. Specifically, we shall see that, when there is a global supersymmetry, there is a conserved trace supersymmetry current with a time-independent trace supercharge Q α , that together with the trace four momentum obeys the Poincaré supersymmetry algebra under the generalized Poisson bracket of Eq. (1.11a). We shall illustrate this statement with three concrete examples, the trace dynamics versions (Adler 1997a,b) of the Wess–Zumino model (Section 3.1), the supersymmetric Yang–Mills model (Section 3.2), and the so-called “matrix model for M theory” (Section 3.3). These three examples are worked out using component field methods; we close in Section 3.4 with a short discussion of a superspace approach, and of the obstruction that prevents the construction of a trace dynamics theory with local supersymmetry. The Wess–Zumino model We begin with the trace dynamics transcription of the Wess–Zumino model.

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In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.

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Journal of Physics A Mathematical and Theoretical•Aug 6, 2014•2 citations
We show that the recently proposed matrix model for M theory obeys the cyclic trace assumptions underlying generalized quantum or trace dynamics. This permits a verification of supersymmetry as an operator calculation, and a calculation of the supercharge density algebra by using the generalized Poisson bracket, in a basis-independent manner that makes no reference to individual matrix elements. Implications for quantization of the model are discussed. Our results are a special case of a general result presented elsewhere, that all rigid supersymmetry theories can be extended to give supersymmetric trace dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a noncommutative trace class graded operator algebra.
In Section 2.4, we illustrated the trace dynamics formalism by constructing the trace dynamics analog of a simple field theory model, in which a Dirac fermion interacts with a scalar Klein–Gordon field. Much of the recent literature in quantum field theory has concerned itself with supersymmetric theories, in which invariance under the Poincaré group has been extended to invariance under the graded Poincaré group, and theories of this type are considered likely to play a central role in the ultimate unification of the forces. Our aim in this chapter (which can be omitted on a first reading) is to show that the trace dynamics formalism naturally extends to globally supersymmetric theories. Specifically, we shall see that, when there is a global supersymmetry, there is a conserved trace supersymmetry current with a time-independent trace supercharge Q α , that together with the trace four momentum obeys the Poincaré supersymmetry algebra under the generalized Poisson bracket of Eq. (1.11a). We shall illustrate this statement with three concrete examples, the trace dynamics versions (Adler 1997a,b) of the Wess–Zumino model (Section 3.1), the supersymmetric Yang–Mills model (Section 3.2), and the so-called “matrix model for M theory” (Section 3.3). These three examples are worked out using component field methods; we close in Section 3.4 with a short discussion of a superspace approach, and of the obstruction that prevents the construction of a trace dynamics theory with local supersymmetry. The Wess–Zumino model We begin with the trace dynamics transcription of the Wess–Zumino model.
In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.
An algebraic view of supersymmetric quantum mechanics is taken, emphasising the couplings between bosonic and fermionic modes in the supercharges. A class of model Hamiltonians is introduced wherein the fermionic (bosonic) operators are canonical and the bosonic (fermionic) ones satisfy a Lie algebra (superalgebra) whose representation theory permits the complete solution of the model in principle. The kinematical symmetry of such models is also described. The examples of one and two bosonic models, with 5U(2) and 5U(3) dynamical algebras respectively, are analysed in detail.
We consider the minimal bosonization realization of supersymmetric shape-invariant systems where generalized supercharge operators are constructed using the partner supersymmetric operators, the parameter potential translation formalism and the reflection operator. We obtain the solution of the eigenvalue equation and study the quantum dynamics of the supersymmetric system including terms in the Hamiltonian which are constructed using the combination of the bosonized supercharge operators. The connections between the bosonized supersymmetric formalism, the Bose–Fermi transformation and the generalization of the -deformed Heisenberg algebra are discussed. As an illustration, we apply the generalized formalism for the case of the trigonometric Rosen–Morse potential.