Dirac Adjoint

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]\displaystyle{ \psi }[/math] be a Dirac spinor. Then its Dirac adjoint is defined as

[math]\displaystyle{ \bar\psi \equiv \psi^\dagger \gamma^0 }[/math]

where [math]\displaystyle{ \psi^\dagger }[/math] denotes the Hermitian adjoint of the spinor [math]\displaystyle{ \psi }[/math], and [math]\displaystyle{ \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]\displaystyle{ \lambda }[/math] is a projective representation of some Lorentz transformation,

[math]\displaystyle{ \psi \mapsto \lambda \psi }[/math],

then, in general,

[math]\displaystyle{ \lambda^\dagger \ne \lambda^{-1} }[/math].

The Hermitian adjoint of a spinor transforms according to

[math]\displaystyle{ \psi^\dagger \mapsto \psi^\dagger \lambda^\dagger }[/math].

Therefore, [math]\displaystyle{ \psi^\dagger\psi }[/math] is not a Lorentz scalar and [math]\displaystyle{ \psi^\dagger\gamma^\mu\psi }[/math] is not even Hermitian.

Dirac adjoints, in contrast, transform according to

[math]\displaystyle{ \bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0 }[/math].

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

[math]\displaystyle{ \bar\psi \mapsto \bar\psi \lambda^{-1} }[/math],

Thus, [math]\displaystyle{ \bar\psi\psi }[/math] transforms as a Lorentz scalar and [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \boldsymbol J = (c \rho, \boldsymbol j) }[/math].

Taking μ = 0 and using the relation for gamma matrices

[math]\displaystyle{ \left(\gamma^0\right)^2 = I }[/math],

the probability density becomes

[math]\displaystyle{ \rho = \psi^\dagger \psi }[/math].

See also

• Dirac equation
• Rarita–Schwinger equation

References

• B. Bransden and C. Joachain (2000). Quantum Mechanics, 2e, Pearson. ISBN:0-582-35691-1.
• M. Peskin and D. Schroeder (1995). An Introduction to Quantum Field Theory, Westview Press. ISBN:0-201-50397-2.
• A. Zee (2003). Quantum Field Theory in a Nutshell, Princeton University Press. ISBN:0-691-01019-6.

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