Dirac adjoint

Short description: Dual to the Dirac spinor

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".

## Contents

* 1 Definition
* 2 Spinors under Lorentz transformations
* 3 Usage
* 4 See also
* 5 References

## Definition

Let ψ be a Dirac spinor. Then its Dirac adjoint is defined as

ψ¯≡ψ†γ0

where ψ† denotes the Hermitian adjoint of the spinor ψ, and γ0 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 λ is a projective representation of some Lorentz transformation,

ψ↦λψ,

then, in general,

λ†≠λ−1.

The Hermitian adjoint of a spinor transforms according to

ψ†↦ψ†λ†.

Therefore, ψ†ψ is not a Lorentz scalar and ψ†γμψ is not even Hermitian.

Dirac adjoints, in contrast, transform according to

ψ¯↦(λψ)†γ0.

Using the identity γ0λ†γ0=λ−1, the transformation reduces to

ψ¯↦ψ¯λ−1,

Thus, ψ¯ψ transforms as a Lorentz scalar and ψ¯γμψ 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

Jμ=cψ¯γμψ

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

J=(cρ,j).

Taking μ = 0 and using the relation for gamma matrices

(γ0)2=I,

the probability density becomes

ρ=ψ†ψ.

## See also

* Dirac equation
* Rarita–Schwinger equation

## References

* B. Bransden; C. Joachain (2000). Quantum Mechanics (2nd ed.). Pearson. ISBN 0-582-35691-1.
* M. Peskin; 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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