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In [[quantum field theory]], the '''Dirac adjoint''' defines the [[dual vector space|dual]] operation of a [[Dirac spinor]].  The Dirac adjoint is motivated by the need to form well-behaved, measurable quantities out of Dirac spinors.  Since the usual [[Hermitian adjoint]] lacks the Lorentz symmetry of the system, the Dirac adjoint must be used instead.


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


== Definition ==
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Let <var>ψ</var> be a [[Dirac spinor]]. Then its Dirac adjoint is defined as
 
:<math>\bar\psi \equiv \psi^\dagger \gamma^0</math>
 
where <var>ψ</var><sup>†</sup> denotes the [[Hermitian adjoint]] of the spinor <var>ψ</var> and <var>γ</var><sup>0</sup> is the time-like [[gamma matrices|gamma matrix]].
 
== Spinors under Lorentz transformations ==
 
The [[Lorentz group]] of [[special relativity]] is not [[compact space|compact]], therefore [[Representations of the Lorentz group|representation]]s of [[Lorentz transformation]]s in the Dirac spinor space are not [[unitary operator|unitary]]. That is, in general,
 
:<math>\lambda^\dagger \ne \lambda^{-1}</math>
 
where <var>λ</var> is the corresponding Lorentz transformation that maps spinors:
 
:<math>\psi \mapsto \lambda \psi</math>.
 
The Hermitian adjoint of spinors transforms according to
 
:<math>\psi^\dagger \mapsto \psi^\dagger \lambda^\dagger</math>.
 
Therefore, using only the Hermitian adjoint, one finds that {{nowrap|<var>ψ</var><sup>†</sup> <var>ψ</var>}} is not a [[Lorentz scalar]] and {{nowrap|<var>ψ</var><sup>†</sup> <var>γ</var><sup><var>μ</var></sup> <var>ψ</var>}} is not even [[self-adjoint operator|Hermitian]].
 
Using the definition, one finds that the Dirac adjoint of spinors transforms according to
 
:<math>\bar\psi \mapsto \left(\lambda \psi\right)^\dagger \gamma^0</math>.
 
Using the identity {{nowrap|<var>γ</var><sup>0</sup> <var>λ</var><sup>†</sup> <var>γ</var><sup>0</sup> {{=}} <var>λ</var><sup>-1</sup>}}, the transformation reduces to
 
:<math>\bar\psi \mapsto \bar\psi \lambda^{-1}</math>,
 
which possesses the required Lorentz symmetry for {{nowrap|{{overline|<var>ψ</var>}} <var>ψ</var>}} and {{nowrap|{{overline|<var>ψ</var>}}  <var>γ</var><sup><var>μ</var></sup> <var>ψ</var>}}.
 
== Usage ==
 
Using the Dirac adjoint, the [[probability current|probability four-current]] <var>'''J'''</var> for a spin-1/2 particle field can be written as
 
:<math>J^\mu = c \bar\psi \gamma^\mu \psi</math>
 
where <var>c</var> is the speed of light and the components of <var>'''J'''</var> represent the probability density <var>ρ</var> and the probability 3-current <var>'''j'''</var>:
 
:<math>\boldsymbol J = (c \rho, \boldsymbol j)</math>.
 
Taking {{nowrap|<var>μ</var> {{=}} 0}} 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. 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.
 
[[Category:Quantum field theory]]
[[Category:Spinors]]
[[Category:Mathematical notation]]

Latest revision as of 01:54, 8 January 2015


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