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6. Spacetime algebra and Dirac theory

Just as it is possible to describe geometric algebra as a fermionic deformed superanalysis it is also possible to describe spacetime algebra in this context. The basis vectors of space-time are the Grassmann elements γ₀, γ₁, γ₂, and γ₃, which fulfill

where we choose here g_μν = diag(1, −1, −1, −1). The corresponding Clifford star product in space-time is

A general supernumber in space-time has the form

where I₄ = γ₀γ₁γ₂γ₃ and only linearly independent terms should appear. With the four-dimensional pseudoscalar I₄ and the Clifford star product (6.2) it is possible to construct analogously to (3.36) the dual basis γ^μ, which gives γ⁰ = γ₀ and γⁱ = −γ_i. Furthermore one can define in analogy to the three-dimensional case a trace:

The Berezin integral acts here again like a projector on the scalar part of F. The definition of the trace by projecting on the scalar part was already given in [19] and it was also stated that the use of geometric algebra greatly simplifys all the trace calculations usually done in the matrix formalism. An explicit expression for the trace can now in the formalism of deformed superanalysis be given by the Berezin integral.

The question is now how a spacetime vector x=x^μγ_μ is related to its space vector part x=xⁱσ_i. In the γ₀-system this can be seen by a space-time split which amounts to star-multiplying by γ₀:

One should note that x=xγ₀=x¹γ₁γ₀+x²γ₂γ₀+x³γ₃γ₀ is a spacetime bivector, but on the other hand it is also a space vector because the two-blades γ_iγ₀ behave like σ_i:

where the four-dimensional star product (6.2) and the three-dimensional star product (3.14) is used in (6.6), as should be clear from the context. The square of the position four vector is x² _C=t²-x² _C.

If a particle is moving in the γ₀-system along x (τ), where τ is the proper time, the proper velocity is given by , with u² _C=1. For the space-time split of the proper velocity one obtains:

Comparing the scalar and the bivector part leads to

and with one gets [3]

It is now also possible to specify a Lorentz transformation from a coordinate system γ_μ to a coordinate system moving in the γ₁-direction. For the coefficients this transformation is given by t = γ (t^′ + βx^′1), x¹ = γ (x^′1 + βt^′), x² = x′², and x³ = x^′3. The condition leads then to

Introducing the angle α so that β = tanh (α) this can be written as

or with as . In general the generators of a passive Lorentz transformation can be calculated with

so that the generators for the boosts and the rotations are

These generators satisfy

and a passive Lorentz transformation is given by

which is a generalization of (6.11).

The Dirac equation can then be written down immediately as [7]

where no slash notation is needed, because one naturally has p=p^μγ_μ. The Wigner function for the Dirac equation is the functional analog of the well-known energy projector of Dirac theory:

Besides the energy one also has the spin as an observable, which is here given by

where s=s^μγ_μ is a vector which fulfills s² _C=-1 and s · p = 0. γ₅ is here γ₅ = iI₄. With and [S_s,p m] _C=0 one sees that the spin Wigner function is given by the functional analog of the spin projector in Dirac theory

and fulfills . The total Wigner function is then the Clifford star product of the two single Wigner functions.

7. Conclusions

There are two formal and conceptual barriers that separate quantum theory from classical theory. The first barrier is that classical theory is described on the phase space while quantum theory is described on the Hilbert space. This conceptual barrier is overcome by the program of deformation quantization that describes quantum theory on the phase space. The second barrier is that one uses in classical mechanics the Gibbs–Heaviside formalism, which cannot take spin into account. In quantum theory where spin is a physical observable it is described in the non-relativistic case by the “Feynman trick”, which substitutes by and in the relativistic case it is introduced by writing p_μγ^μ. Both notations clearly indicate that the σ_i and the γ_μ are basis vectors, but this is obscured by representing them by matrices. The work of Hestenes has clarified this point by formulating classical and quantum theory in the same formalism of geometric algebra. The surprising thing is now that also this second barrier can be overcome in terms of the star product formalism, so that classical and quantum theory can be unified on a formal level. Both can be described by the formalism of deformed superanalysis, where classical mechanics is a “half-deformed” formalism, that means the deformation only takes place in the Grassmann sector of superanalysis, while quantum mechanics leads to a “totally deformed” formalism, where also the product of the scalar coefficients are deformed. This shows on a formal level that quantum theory is more fundamental than classical theory.

The star product formalism has also advantages in the context of geometric calculus, because it gives an explicit expression for the geometric product. Geometric algebra, in the way Hestenes constructed it, is formulated with respect to the scalar and the wedge product, which represent the lowest and the highest order terms of the geometric product. All other terms of the geometric product are then formulated with the help of these two products. This approach is very practical, especially if one has only terms that are at most bivectors. But in the general case the highest and the lowest terms of an expansion have on a formal level the same status as all other terms. The star product gives now all these terms of different grade as terms of an expansion, that can be calculated in a straightforward fashion.

SLE for theoretical physicists

John Cardy ^,

Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK
All Souls College, Oxford, UK

Received 14 March 2005; accepted 4 April 2005. Available online 12 May 2005.

Abstract

This article provides an introduction to Schramm (stochastic)–Loewner evolution (SLE) and to its connection with conformal field theory, from the point of view of its application to two-dimensional critical behaviour. The emphasis is on the conceptual ideas rather than rigorous proofs.

Article Outline

1. Introduction

1.1. Historical overview

1.2. Aims of this article

2. Random curves and lattice models

2.1. The Ising and percolation models

2.1.1. Exploration process

2.2. O (n) model

2.3. Potts model

2.4. Coulomb gas methods

2.4.1. Winding angle distribution

2.4.2. N-leg exponent

3. SLE

3.1. The postulates of SLE

3.2. Loewner’s equation

3.3. Schramm–Loewner evolution

3.4. Simple properties of SLE

3.4.1. Phases of SLE

3.4.2. SLE duality

3.5. Special values of κ

3.5.1. Locality

3.5.2. Restriction

3.6. Radial SLE and the winding angle

3.6.1. Identification with lattice models

4. Calculating with SLE

4.1. Schramm’s formula

4.2. Crossing probability

Характеристики

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