Definition
In theoretical physics, a Dirac bracket is a bilinear operation on functions of phase‑space variables that generalizes the Poisson bracket to systems with constraints. It is defined so that the brackets of any function with the constraints vanish, thereby allowing the constraints to be treated as strong (i.e., identically zero) equations after quantization.
Historical Background
The concept was introduced by Paul A. M. Dirac in his seminal works on constrained Hamiltonian dynamics, notably in Lectures on Quantum Mechanics (1964) and Generalized Hamiltonian Dynamics (1967). Dirac developed the bracket to provide a systematic procedure for quantizing systems whose Lagrangian formulation leads to dependent canonical coordinates, such as gauge theories and relativistic particles.
Mathematical Formulation
Consider a phase space with coordinates $(q_i, p_i)$ and a set of second‑class constraints $\phi_a(q,p)=0$ (where the constraint matrix $C_{ab}={\phi_a,\phi_b}$ is invertible). The Dirac bracket of two phase‑space functions $F$ and $G$ is defined as
$$ {F, G}_D ;=; {F, G};-;{F, \phi_a},C^{ab},{\phi_b, G}, $$
where ${, , ,}$ denotes the usual Poisson bracket, $C^{ab}$ is the inverse of $C_{ab}$, and summation over repeated indices $a,b$ is implied.
Key properties:
- Antisymmetry: ${F,G}_D = -{G,F}_D$.
- Jacobi identity: The Dirac bracket satisfies the Jacobi identity, making it a legitimate Lie bracket on the reduced phase space.
- Compatibility with constraints: For any constraint $\phi_a$, ${\phi_a, F}_D = 0$ for all functions $F$. Consequently, the constraints can be imposed strongly (i.e., set to zero before computing further brackets).
- Reduction to Poisson bracket: If no second‑class constraints are present (or if the constraint matrix is singular), the Dirac bracket reduces to the ordinary Poisson bracket.
Relation to Quantization
In the canonical quantization of constrained systems, Dirac proposed replacing Poisson brackets by Dirac brackets before promoting them to commutators:
$$ {F, G}_D ;\longrightarrow; \frac{1}{i\hbar}[ \hat{F}, \hat{G} ] . $$
This procedure ensures that the quantum operators respect the constraints at the operator level.
Applications
| Field | Typical Use of Dirac Brackets |
|---|---|
| Gauge Theories | Treatment of gauge‑fixing conditions as second‑class constraints (e.g., in electrodynamics and non‑abelian Yang‑Mills theory). |
| General Relativity | Hamiltonian formulation of the ADM formalism and analysis of diffeomorphism constraints. |
| Relativistic Particles & Strings | Quantization of the relativistic particle, Nambu–Goto string, and related models where re‑parameterization invariance leads to constraints. |
| Constrained Mechanical Systems | Systems with holonomic or non‑holonomic constraints, such as the motion of a rigid body with attached sliders. |
| BRST Formalism | Dirac brackets serve as a precursor to the construction of the BRST charge for systems with gauge symmetry. |
Examples
-
Particle on a Sphere
For a particle constrained to the surface $x^2 + y^2 + z^2 = R^2$, the primary constraint $\phi_1 = x^2 + y^2 + z^2 - R^2$ and the secondary constraint $\phi_2 = \mathbf{p}\cdot \mathbf{x}$ are second‑class. The Dirac bracket modifies the canonical brackets to enforce the spherical constraint at the level of dynamics. -
Electromagnetic Field in Coulomb Gauge
Imposing the Coulomb gauge condition $ abla\cdot\mathbf{A}=0$ introduces a set of second‑class constraints. The resulting Dirac brackets yield the transverse commutation relations for the vector potential, eliminating unphysical longitudinal modes.
Key References
- P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University (1964).
- P. A. M. Dirac, Generalized Hamiltonian Dynamics, Canadian Journal of Mathematics 2 (1950) 129–148.
- M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press (1992).
- L. D. Faddeev and R. Jackiw, “Hamiltonian reduction of unconstrained and constrained systems,” Physical Review Letters 60 (1988) 1692.
See Also
- Poisson bracket
- Constraint (classical mechanics)
- Second‑class constraint
- BRST quantization
- Symplectic geometry
This entry summarizes the established concept of the Dirac bracket as used in classical and quantum theories of constrained systems.
