40 Minkowski Space and the Lorentz Group

For the case of non-relativistic quantum mechanics, we saw that systems with an arbitrary number of particles, bosons or fermions, could be described by taking as dual phase space the state space ${\mathcal{H}}_1$ of the single-particle quantum theory. This space is infinite dimensional, but it is linear and it can be quantized using the same techniques that work for the finite dimensional harmonic oscillator. This is an example of a quantum field theory since it is a space of functions that is being quantized.

We would like to find some similar way to proceed for the case of relativistic systems, finding relativistic quantum field theories capable of describing arbitrary numbers of particles, with the energy-momentum relationship $E^{\bar{2}}=|\mathbf{p}{|}^2{c}^2+m^2{c}^4$ characteristic of special relativity, not the non-relativistic limit |p| ≪ mc where $\begin{array}{r}E=\frac{{|\mathbf{p}|}^2}{2m}\end{array}$ . In general, a phase space can be thought of as the space of initial conditions for an equation of motion, or equivalently, as the space of solutions of the equation of motion. In the non-relativistic field theory, the equation of motion is the first-order in time Schr¨odinger equation, and the phase space is the space of fields (wavefunctions) at a specified initial time, say $t=0$ . This space carries a representation of the time-translation group $R$ and the Euclidean group $E(3)={\mathbf{R}}^3⋊SO(3)$ . To construct a relativistic quantum field theory, we want to find an analog of this space of wavefunctions. It will be some sort of linear space of functions satisfying an equation of motion, and we will then quantize by applying harmonic oscillator methods.

Just as in the non-relativistic case, the space of solutions to the equation of motion provides a representation of the group of space-time symmetries of the theory. This group will now be the Poincar´e group, a ten dimensional group which includes a four dimensional subgroup of translations in space-time, and a six dimensional subgroup (the Lorentz group), which combines spatial rotations and “boosts” (transformations mixing spatial and time coordinates). The representation of the Poincar´e group on the solutions to the relativistic wave equation will in general be reducible. Irreducible such representations will be the objects corresponding to elementary particles. This chapter will deal with the Lorentz group itself, chapter 41 with its representations, and chapter 42 will move on to the Poincar´e group and its representations.

中文目录

Chapter contents

• 40.1 Minkowski space
• 40.2 The Lorentz group and its Lie algebra
• 40.3 The Fourier transform in Minkowski space
• 40.4 Spin and the Lorentz group
• 40.5 For further reading

Previous chapter: 39 Quantization of Infinite dimensional Phase Spaces

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