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Since the Dirac equation was originally invented to describe the electron, we will generally speak of "electrons" in this article. Actually, the equation also applies to Quark s, which are also elementary spin-½ particles. A modified Dirac equation can be used to approximately describe Proton s and Neutron s, which are not elementary particles (they are made up of quarks). Another modification of the Dirac equation, called the Majorana Equation , is thought to describe Neutrino s.

The Dirac equation is

: \left(\alpha_0 mc^2 + \sum_{j = 1}^3 \alpha_j p_j \, c ight) \psi (\mathbf{x},t) = i \hbar rac{\partial\psi}{\partial t} (\mathbf{x},t)

where ''m'' is the Rest Mass of the electron, ''c'' is the Speed Of Light , ''p'' is the Momentum operator, \hbar is the reduced Planck's Constant , x and ''t'' are the Space and Time coordinates respectively, and ''ψ''(x, ''t'') is a four-component Wavefunction . (The wavefunction has to be formulated as a four-component Spinor , rather than a simple Scalar , due to the demands of special relativity. The physical meanings of the components are discussed below.)

The α's are Linear Operators that act on the wavefunction. Their most fundamental property is that they must anticommute with each other. In other words,
:\alpha_i\alpha_j = -\alpha_j\alpha_i,
where i
e j, and i and j range from zero to three. The simplest way to obtain such properties is with 4×4 Matrices . There is no set of matrices of smaller dimension fulfilling the anticommutation requirements. That four dimensional matrices are necessary turns out to have physical significance.

A convenient (but not unique) choice of \alphas is

:\alpha_0 = \begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1 \end{bmatrix} \quad \alpha_1 = \begin{bmatrix} 0 & 0 & 0 & 1 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 1 & 0 & 0 & 0 \end{bmatrix} ,

:\alpha_2 = \begin{bmatrix} 0 & 0 & 0 & -i \ 0 & 0 & i & 0 \ 0 & -i& 0 & 0 \ i & 0 & 0 & 0 \end{bmatrix} \quad \alpha_3 = \begin{bmatrix} 0 & 0 & 1 & 0 \ 0 & 0 & 0 & -1 \ 1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \end{bmatrix} ,

known as Dirac Matrices . All possible choices are related by Similarity Transformation s because Dirac spinors are unique Representation Theoretically .

The Dirac equation describes the Probability Amplitude s for a ''single'' electron. This is a single-particle theory, in other words it does not account for the creation and destruction of the particles. It gives a good prediction of the magnetic moment of the electron and explains much of the Fine Structure observed in Atom ic Spectral Line s. It also explains the spin of the electron. Two of the four solutions of the equation correspond to the two spin states of the electron. The other two solutions make the peculiar prediction that there exist an infinite set of quantum states in which the electron possesses negative Energy . This strange result led Dirac to predict, via a remarkable hypothesis known as "hole theory", the existence of particles behaving like positively-charged electrons. This prediction was verified by the discovery of the Positron in 1932 .

Despite these successes, the theory is flawed by its neglect of the possibility of creating and destroying particles, one of the basic consequences of relativity. This difficulty is resolved by reformulating it as a Quantum Field Theory . Adding a quantized Electromagnetic Field to this theory leads to the theory of Quantum Electrodynamics (QED). Moreover the equation cannot fully account for particles of negative energy but is restricted to positive energy particles.

A similar equation for spin 3/2 particles is called the Rarita-Schwinger Equation .


DERIVATION OF THE DIRAC EQUATION


The Dirac equation is a relativistic extension of the Schrödinger Equation , which describes the time-evolution of a quantum mechanical system:



where ''w'' is a constant four-component spinor and ''p'' is the momentum of the particle, as we can verify by applying the momentum operator to this wavefunction. In the Dirac representation, the equation for ''ψ0'' reduces to the Eigenvalue Equation :

: \begin{bmatrix} mc^2 & 0 & pc & 0 \ 0 & mc^2 & 0 & -pc \ pc & 0 & -mc^2 & 0 \ 0 & -pc & 0 & -mc^2 \end{bmatrix} w = E w

For each value of ''p'', there are two eigenspaces, both two-dimensional. One eigenspace contains positive eigenvalues, and the other negative eigenvalues, of the form:

:E_\pm (p) = \pm \sqrt{(mc^2)^2 + (pc)^2}

The positive eigenspace is spanned by the eigenstates:

:\left\{ \begin{bmatrix}pc \ 0 \ \epsilon \ 0 \end{bmatrix} \,,\, \begin{bmatrix}0 \ pc \ 0 \ - \epsilon \end{bmatrix} ight\} imes rac{1}{\sqrt{\epsilon^2+(pc)^2}}

and the negative eigenspace by the eigenstates:

:\left\{ \begin{bmatrix}-\epsilon \ 0 \ pc \ 0 \end{bmatrix} \,,\, \begin{bmatrix}0 \ \epsilon \ 0 \ pc \end{bmatrix} ight\} imes rac{1}{\sqrt{\epsilon^2+(pc)^2}}

where

  :<math> \left( Rac{1}{2m} \sum J p J - E A J(\mathbf{x}, T)^2 - Rac{\hbar E}{2mc} \sum J \sigma J B J(\mathbf{x}) Ight) \begin{bmatrix}\psi 1 \ \psi 2 \end{bmatrix}</math><br>&nbsp<math> (E - mc^2) \begin{bmatrix}\psi_1 \ \psi_2 \end{bmatrix}</math>


where ''H''free is the Dirac Hamiltonian for a free electron and ''H''int is the Hamiltonian of the electromagnetic interaction. The latter may be written as

:H_{\mathrm{int}} = e arphi(\mathbf{x}, t) - ec \sum_{j=1}^3 \alpha_j A_j(\mathbf{x}, t)

It has the Expected Value

:\langle H angle = \int \, \psi^\dagger H_{\mathrm{int}} \psi \, d^3x = \int \, \left( ho arphi - \sum_{i=1}^3 j_i A_i ight) \, d^3x

where ''ρ'' is the electric charge density and j is the electric current density defined earlier. The integrand in the final expression is the interaction energy density. It is a relativistically covariant scalar quantity, as we can see by writing it in terms of the current-charge Four-vector ''j'' = (''ρc'',j) and the potential four-vector ''A'' = (''φ/c'','''A'''):

:\langle H angle = \int \, \left( \sum_{\mu,
u = 0}^3 \eta^{\mu
u} j_\mu A_
u ight) \; d^3r

where ''η'' is the Metric of Flat Spacetime :

:\eta^{00} = 1
:\eta^{ii} \;= -1 \quad\, orall \, i=1,2,3
:\eta^{\mu
u} = 0 \qquad orall \, \mu
e
u


RELATIVISTICALLY COVARIANT NOTATION


Let us return to the Dirac equation for the free electron. It is often useful to write the equation in a relativistically covariant form, in which the derivatives with time and space are treated on the same footing.

To do this, first recall that the momentum operator p acts like a spatial derivative:

:\mathbf{p} \psi(\mathbf{x},t) = - i \hbar
abla \psi(\mathbf{x},t)

Multiplying each side of the Dirac equation by ''α0'' (recalling that ''α0²=I'') and plugging in the above definition of p, we obtain

: \left i\hbar c \left(\alpha_0 rac{\partial}{c \partial t} + \sum_{j=1}^3 \alpha_0 \alpha_j rac{\partial}{\partial x_j} ight) - mc^2 ight \psi = 0

Now, define four Gamma Matrices :

: \gamma^0 \equiv \alpha_0 \,,\quad \gamma^j \equiv \alpha_0 \alpha_j

These matrices possess the property that

:\left\{\gamma^\mu , \gamma^
u ight\} = 2\eta^{\mu
u} \cdot I\,,\quad \mu,
u = 0, 1, 2, 3

where ''η'' once again stands for the metric of flat spacetime. These relations define a Clifford Algebra called the Dirac algebra.

The Dirac equation may now be written, using the position-time four-vector ''x'' = (''ct'',x), as

:\left(i\hbar c \, \sum_{\mu=0}^3 \; \gamma^\mu \, \partial_\mu - mc^2 ight) \psi = 0

With this notation, the Dirac equation can be generated by extremising the Action

:\mathcal{S} = \int \bar\psi(i \hbar c \, \sum_\mu \gamma^\mu \partial_\mu - mc^2)\psi \, d^4 x

where

:\bar\psi \equiv \psi^\dagger \gamma_0

is called the Dirac Adjoint of ''ψ''. This is the basis for the use of the Dirac equation in Quantum Field Theory .

A notation called the " Feynman Slash " is sometimes used. Writing

:a\!\!\!/ \leftrightarrow \sum_\mu \gamma^\mu a_\mu

the Dirac equation becomes

:(i \hbar c \, \partial\!\!\!/ - mc^2) \psi = 0

and the expression for the action becomes

:\mathcal{S} = \int \bar\psi(i \hbar c \, \partial \!\!\!/ - mc^2)\psi \, d^4 x

In this notation electromagnetic interaction can be added simply by promoting the partial derivative to Gauge Covariant Derivative :

:\partial_\mu ightarrow D_\mu = \partial_\mu - i e A_\mu


DIRAC BILINEARS

There are five different (neutral) Dirac bilinear terms not involving any derivatives:



A Dirac mass term is an S coupling. A Yukawa coupling may be S or P. The electromagnetic coupling is V. The weak interactions are V-A.


SEE ALSO




REFERENCES



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