Quantizing a field theory involves expressing the theory in terms of operators acting on a Hilbert space, where the fields are promoted to operator-valued objects. The quantization procedure aims to replace classical fields with quantum fields and to define appropriate commutation or anticommutation relations for these fields.
There are two main methods used for quantizing field theories: canonical quantization and path integral quantization. Let's discuss each method and highlight their differences.
Canonical Quantization: Canonical quantization follows a similar procedure to quantizing particles in quantum mechanics, but now extended to fields. Here's a step-by-step overview of the canonical quantization process for a field theory:
a. Field expansion: The classical field is expanded into Fourier modes or other appropriate basis functions. b. Canonical momentum: The conjugate momentum field is defined as the derivative of the Lagrangian with respect to the time derivative of the field. c. Canonical commutation relations: The fields and their conjugate momenta are promoted to operators, and commutation relations (or anticommutation relations for fermionic fields) are imposed on these operator-valued fields. These commutation relations are typically taken from the equal-time commutation relations of the corresponding quantum mechanical operators. d. Field quantization: The fields are expanded in terms of creation and annihilation operators, which are related to the Fourier modes of the fields. These operators act on a Fock space, which represents the quantum state space. e. Hamiltonian operator: The classical Hamiltonian is replaced by a quantum operator, usually expressed in terms of the field operators and their conjugate momenta. f. Time evolution: The time evolution of the field operators is determined by the Heisenberg equations of motion, derived from the Hamiltonian operator.
Path Integral Quantization: Path integral quantization, also known as the Feynman path integral approach, takes a different approach to quantization. Instead of working with operators and commutation relations directly, it expresses transition amplitudes as integrals over all possible field configurations. Here's a brief outline of the path integral quantization procedure:
a. Functional integral: The transition amplitude between initial and final field configurations is expressed as a path integral over all field configurations in between. b. Action functional: The action functional, which depends on the fields and their derivatives, is exponentiated and integrated over all field configurations. c. Stationary phase approximation: The path integral is evaluated using the stationary phase approximation, where the dominant contributions come from the field configurations that extremize the action. d. Generating functional: The path integral can be used to construct a generating functional, which generates all correlation functions of the field theory. e. Effective action: The path integral is related to the effective action, which encodes the quantum corrections to the classical action of the theory.
Differences between the methods:
- Canonical quantization is based on promoting classical fields to operator-valued fields, while path integral quantization expresses transition amplitudes as integrals over all field configurations.
- In canonical quantization, the Hilbert space is constructed by expanding the fields in terms of creation and annihilation operators, while path integral quantization does not directly construct a Hilbert space.
- Canonical quantization explicitly deals with operators and commutation relations, while path integral quantization works with functional integrals.
- Canonical quantization often provides a more direct connection to the operator formalism and is well-suited for perturbative calculations, while path integral quantization offers a more intuitive and diagrammatic approach to calculations, particularly for scattering amplitudes.
Both canonical and path integral quantization have their strengths and are complementary in many ways. The choice between the methods depends on the specific problem at hand, the symmetries involved, and the preferences of the physicist.