Английская Википедия:Dirac adjoint

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Шаблон:Short description In quantum field theory, the Dirac adjoint defines the dual operation of a Dirac spinor. The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors, replacing the usual role of the Hermitian adjoint.

Possibly to avoid confusion with the usual Hermitian adjoint, some textbooks do not provide a name for the Dirac adjoint but simply call it "ψ-bar".

Definition

Let <math>\psi</math> be a Dirac spinor. Then its Dirac adjoint is defined as

<math>\bar\psi \equiv \psi^\dagger \gamma^0</math>

where <math>\psi^\dagger</math> denotes the Hermitian adjoint of the spinor <math>\psi</math>, and <math>\gamma^0</math> is the time-like gamma matrix.

Spinors under Lorentz transformations

The Lorentz group of special relativity is not compact, therefore spinor representations of Lorentz transformations are generally not unitary. That is, if <math>\lambda</math> is a projective representation of some Lorentz transformation,

<math>\psi \mapsto \lambda \psi</math>,

then, in general,

<math>\lambda^\dagger \ne \lambda^{-1}</math>.

The Hermitian adjoint of a spinor transforms according to

<math>\psi^\dagger \mapsto \psi^\dagger \lambda^\dagger</math>.

Therefore, <math>\psi^\dagger\psi</math> is not a Lorentz scalar and <math>\psi^\dagger\gamma^\mu\psi</math> is not even Hermitian.

Dirac adjoints, in contrast, transform according to

<math>\bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0</math>.

Using the identity <math>\gamma^0 \lambda^\dagger \gamma^0 = \lambda^{-1}</math>, the transformation reduces to

<math>\bar\psi \mapsto \bar\psi \lambda^{-1}</math>,

Thus, <math>\bar\psi\psi</math> transforms as a Lorentz scalar and <math>\bar\psi\gamma^\mu\psi</math> as a four-vector.

Usage

Using the Dirac adjoint, the probability four-current J for a spin-1/2 particle field can be written as

<math>J^\mu = c \bar\psi \gamma^\mu \psi</math>

where c is the speed of light and the components of J represent the probability density ρ and the probability 3-current j:

<math>\boldsymbol J = (c \rho, \boldsymbol j)</math>.

Taking Шаблон:Nowrap and using the relation for gamma matrices

<math>\left(\gamma^0\right)^2 = I</math>,

the probability density becomes

<math>\rho = \psi^\dagger \psi</math>.

See also

• Dirac equation
• Rarita–Schwinger equation

References

• B. Bransden and C. Joachain (2000). Quantum Mechanics, 2e, Pearson. Шаблон:ISBN.
• M. Peskin and D. Schroeder (1995). An Introduction to Quantum Field Theory, Westview Press. Шаблон:ISBN.
• A. Zee (2003). Quantum Field Theory in a Nutshell, Princeton University Press. Шаблон:ISBN.