43 The Klein-Gordon Equation and Scalar Quantum Fields

In the non-relativistic case we found that it was possible to build a quantum theory describing arbitrary numbers of particles by taking as dual phase space $M$ the single-particle space ${\mathcal{H}}_1$ of solutions to the free particle Schr¨odinger equation. To get the same sort of construction for relativistic systems, one possibility is to take as dual phase space the space of solutions of a relativistic wave equation known as the Klein-Gordon equation.

A major diference with the non-relativistic case is that the equation of motion is second-order in time, so to parametrize solutions in ${\mathcal{H}}_1$ one needs not just the wavefunction at a fixed time, but also its time derivative. In addition, consistency with conditions of causality and positive energy of states requires making a very diferent choice of complex structure ${}^{J,}$ one that is not the complex structure coming from the complex-valued nature of the wavefunction (a choice which may in any case be unavailable, since in the simplest theory Klein-Gordon wavefunctions will be real-valued). In the relativistic case, an appropriate complex structure $J_r$ is defined by complexifying the space of solutions, and then taking $J_r$ to have value +$i$ on positive energy solutions and −$i$ on negative energy solutions. This implies a diferent physical interpretation than in the non-relativistic case, with a non-negative energy assignment to states achieved by interpreting negative energy solutions as corresponding to positive energy antiparticle states moving backwards in time.

中文目录

Chapter contents

• 43.1 The Klein-Gordon equation and its solutions
• 43.2 The symplectic and complex structures on M
• 43.3 Hamiltonian and dynamics of the Klein-Gordon theory
• 43.4 Quantization of the Klein-Gordon theory
• 43.5 The scalar field propagator
• 43.6 Interacting scalar field theories: some comments
• 43.7 For further reading

Previous chapter: 42 The Poincaré Group and its Representations

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