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Register a new userCalculus, Relativity and Non-Commutative Worlds
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Authors
Louis H Kauffman
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This paper shows how gauge theoretic structures arise naturally in a non-commutative calculus. Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry. We show how a covariant version of the Levi-Civita connection arises naturally in this commutator calculus. This connection satisfies the formula $$Gamma_{kij} + Gamma_{ikj} = abla_{j}g_{ik} = partial_{j} g_{ik} + [g_{ik}, A_j].$$ and so is exactly a generalization of the connection defined by Hermann Weyl in his original gauge theory. In the non-commutative world $cal N$ the metric indeed has a wider variability than the classical metric and its angular holonomy. Weyls idea was to work with such a wider variability of the metric. The present formalism provides a new context for Weyls original idea.
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We discuss a non--commutative integration calculus arising in the mathematical description of anomalies in fermion--Yang--Mills systems. We consider the differential complex of forms $u_0ccr{eps}{u_1}cdotsccr{eps}{u_n}$ with $eps$ a grading operator on a Hilbert space $cH$ and $u_i$ bounded operators on $cH$ which naturally contains the compactly supported de Rham forms on $R^d$ (i.e. $eps$ is the sign of the free Dirac operator on $R^d$ and $cH$ a $L^2$--space on $R^d$). We present an elementary proof that the integral of $d$--forms $int_{R^d}trac{X_0dd X_1cdots dd X_d}$ for $X_iinMap(R^d;gl_N)$, is equal, up to a constant, to the conditional Hilbert space trace of $Gamma X_0ccr{eps}{X_1}cdotsccr{eps}{X_d}$ where $Gamma=1$ for $d$ odd and $Gamma=gamma_{d+1}$ (`$gamma_5$--matrix) a spin matrix anticommuting with $eps$ for $d$ even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes non--commutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace.
This chapter is divided into two parts. The first is largely expository and builds on Karandikars axiomatisation of It{^o} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{^o} and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the It{^o} stochastic exponential of matrix-valued semimartingales.
We show that it is in principle possible to construct dualities between commutative and non-commutative theories in a systematic way. This construction exploits a generalization of the exact renormalization group equation (ERG). We apply this to the simple case of the Landau problem and then generalize it to the free and interacting non-canonical scalar field theory. This constructive approach offers the advantage of tracking the implementation of the Lorentz symmetry in the non-commutative dual theory. In principle, it allows for the construction of completely consistent non-commutative and non-local theories where the Lorentz symmetry and unitarity are still respected, but may be implemented in a highly non-trivial and non-local manner.
The Chevalley-Eilenberg differential calculus and differential operators over N-graded commutative rings are constructed. This is a straightforward generalization of the differential calculus over commutative rings, and it is the most general case of the differential calculus over rings that is not the non-commutative geometry. Since any N-graded ring possesses the associated Z_2-graded structure, this also is the case of the graded differential calculus over Grassmann algebras and the supergeometry and field theory on graded manifolds.
The effect of non-commutativity on electromagnetic waves violates Lorentz invariance: in the presence of a background magnetic induction field b, the velocity for propagation transverse to b differs from c, while propagation along b is unchanged. In principle, this allows a test by the Michelson-Morley interference method. We also study non-commutativity in another context, by constructing the theory describing a charged fluid in a strong magnetic field, which forces the fluid particles into their lowest Landau level and renders the fluid dynamics non-commutative, with a Moyal product determined by the background magnetic field.
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