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| In [[quantum field theory]], the '''Dirac spinor''' is the [[bispinor]] in the [[Plane wave|plane-wave]] solution
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| :<math>\psi = \omega_\vec{p}\;e^{-ipx} \;</math>
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| of the free [[Dirac equation]],
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| :<math>(i\gamma^\mu\partial_{\mu}-m)\psi=0 \;,</math>
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| where (in the units <math>\scriptstyle c \,=\, \hbar \,=\, 1</math>)
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| :<math>\scriptstyle\psi</math> is a [[Theory of relativity|relativistic]] [[spin-1/2]] [[Field (physics)|field]],
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| :<math>\scriptstyle\omega_\vec{p}</math> is the Dirac [[spinor]] related to a plane-wave with [[wave-vector]] <math>\scriptstyle\vec{p}</math>,
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| :<math>\scriptstyle px \;\equiv\; p_\mu x^\mu</math>,
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| :<math>\scriptstyle p^\mu \;=\; \{\pm\sqrt{m^2+\vec{p}^2},\, \vec{p}\}</math> is the four-wave-vector of the plane wave, where <math>\scriptstyle\vec{p}</math> is arbitrary,
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| :<math>\scriptstyle x^\mu</math> are the four-coordinates in a given [[inertial frame]] of reference.
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|
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| The Dirac spinor for the positive-frequency solution can be written as
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| :<math>
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| \omega_\vec{p} =
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| \begin{bmatrix}
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| \phi \\ \frac{\vec{\sigma}\vec{p}}{E_{\vec{p}} + m} \phi
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| \end{bmatrix} \;,
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| </math>
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| where
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| :<math>\scriptstyle\phi</math> is an arbitrary two-spinor,
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| :<math>\scriptstyle\vec{\sigma}</math> are the [[Pauli matrices]],
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| :<math>\scriptstyle E_\vec{p}</math> is the positive square root <math>\scriptstyle E_{\vec{p}} \;=\; +\sqrt{m^2+\vec{p}^2}</math>
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|
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| ==Derivation from Dirac equation==
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| The Dirac equation has the form
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| :<math>\left(-i \vec{\alpha} \cdot \vec{\nabla} + \beta m \right) \psi = i \frac{\partial \psi}{\partial t} \,</math>
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| In order to derive the form of the four-spinor <math>\scriptstyle\omega</math> we have to first note the value of the matrices α and β:
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| :<math>\vec\alpha = \begin{bmatrix} \mathbf{0} & \vec{\sigma} \\ \vec{\sigma} & \mathbf{0} \end{bmatrix} \quad \quad \beta = \begin{bmatrix} \mathbf{I} & \mathbf{0} \\ \mathbf{0} & -\mathbf{I} \end{bmatrix} \,</math>
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| These two 4×4 matrices are related to the [[Gamma matrices|Dirac gamma matrices]]. Note that '''0''' and '''I''' are 2×2 matrices here.
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| The next step is to look for solutions of the form
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| :<math>\psi = \omega e^{-i p \cdot x}</math>,
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| while at the same time splitting ω into two two-spinors:
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| :<math>\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} \,</math>.
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| ===Results===
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| Using all of the above information to plug into the Dirac equation results in
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| :<math>E \begin{bmatrix} \phi \\ \chi \end{bmatrix} =
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| \begin{bmatrix} m \mathbf{I} & \vec{\sigma}\vec{p} \\ \vec{\sigma}\vec{p} & -m \mathbf{I} \end{bmatrix} \begin{bmatrix} \phi \\ \chi \end{bmatrix} \,</math>.
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| This matrix equation is really two coupled equations:
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| :<math>\left(E - m \right) \phi = \left(\vec{\sigma}\vec{p} \right) \chi \,</math>
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| :<math>\left(E + m \right) \chi = \left(\vec{\sigma}\vec{p} \right) \phi \,</math>
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| Solve the 2nd equation for <math>\scriptstyle \chi \,</math> and one obtains
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| :<math>\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} = \begin{bmatrix} \phi \\ \frac{\vec{\sigma}\vec{p}}{E + m} \phi \end{bmatrix} \,</math>.
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| Solve the 1st equation for <math>\phi \,</math> and one finds
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| :<math>\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} = \begin{bmatrix} - \frac{\vec{\sigma}\vec{p}}{-E + m} \chi \\ \chi \end{bmatrix} \,</math>.
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| This solution is useful for showing the relation between [[anti-particle]] and particle.
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| ==Details==
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| ===Two-spinors===
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| The most convenient definitions for the two-spinors are:
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| :<math>\phi^1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad \quad \phi^2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,</math>
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| and
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| :<math>\chi^1 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \quad \chi^2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \,</math>
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| ===Pauli matrices===
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| The [[Pauli matrices]] are
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| :<math> | |
| \sigma_1 =
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| \begin{bmatrix}
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| 0&1\\
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| 1&0
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| \end{bmatrix}
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| \quad \quad
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| \sigma_2 =
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| \begin{bmatrix}
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| 0&-i\\
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| i&0
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| \end{bmatrix}
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| \quad \quad
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| \sigma_3 =
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| \begin{bmatrix}
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| 1&0\\
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| 0&-1
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| \end{bmatrix}
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| </math>
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| Using these, one can calculate:
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| :<math>\vec{\sigma}\vec{p} = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3 =
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| \begin{bmatrix}
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| p_3 & p_1 - i p_2 \\
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| p_1 + i p_2 & - p_3
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| \end{bmatrix}</math>
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| ==Four-spinor for particles==
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| Particles are defined as having ''positive'' energy. The normalization for the four-spinor ω is chosen so that <math>\scriptstyle\omega^\dagger \omega \;=\; 2 E \,</math> {{Elucidate|date=February 2012}}. These spinors are denoted as ''u'':
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| :<math> u(\vec{p}, s) = \sqrt{E+m}
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| \begin{bmatrix}
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| \phi^{(s)}\\
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| \frac{\vec{\sigma} \cdot \vec{p} }{E+m} \phi^{(s)}
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| \end{bmatrix} \,</math>
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| where ''s'' = 1 or 2 (spin "up" or "down")
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| Explicitly,
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| :<math>u(\vec{p}, 1) = \sqrt{E+m} \begin{bmatrix}
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| 1\\
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| 0\\
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| \frac{p_3}{E+m} \\
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| \frac{p_1 + i p_2}{E+m}
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| \end{bmatrix} \quad \mathrm{and} \quad
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| u(\vec{p}, 2) = \sqrt{E+m} \begin{bmatrix}
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| 0\\
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| 1\\
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| \frac{p_1 - i p_2}{E+m} \\
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| \frac{-p_3}{E+m}
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| \end{bmatrix} </math>
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| ==Four-spinor for anti-particles==
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| Anti-particles having ''positive'' energy <math>\scriptstyle E</math> are defined as particles having ''negative'' energy and propagating backward in time. Hence changing the sign of <math>\scriptstyle E</math> and <math>\scriptstyle \vec{p}</math> in the four-spinor for particles will give the four-spinor for anti-particles:
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| :<math> v(\vec{p},s) = \sqrt{E+m}
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| \begin{bmatrix}
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| \frac{\vec{\sigma} \cdot \vec{p} }{E+m} \chi^{(s)}\\
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| \chi^{(s)}
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| \end{bmatrix} \,</math>
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| Here we choose the <math>\scriptstyle\chi</math> solutions. Explicitly,
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| :<math>v(\vec{p}, 1) = \sqrt{E+m} \begin{bmatrix}
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| \frac{p_1 - i p_2}{E+m} \\
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| \frac{-p_3}{E+m} \\
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| 0\\
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| 1
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| \end{bmatrix} \quad \mathrm{and} \quad
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| v(\vec{p}, 2) = \sqrt{E+m} \begin{bmatrix}
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| \frac{p_3}{E+m} \\
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| \frac{p_1 + i p_2}{E+m} \\
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| 1\\
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| 0\\
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| \end{bmatrix} </math>
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| ==Completeness relations==
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| The completeness relations for the four-spinors ''u'' and ''v'' are
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| :<math>\sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = p\!\!\!/ + m \,</math>
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| :<math>\sum_{s=1,2}{v^{(s)}_p \bar{v}^{(s)}_p} = p\!\!\!/ - m \,</math>
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| where
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| :<math>p\!\!\!/ = \gamma^\mu p_\mu \,</math> (see [[Feynman slash notation#With four-momentum|Feynman slash notation]])
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| :<math>\bar{u} = u^{\dagger} \gamma^0 \,</math>
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| ==Dirac spinors and the Dirac algebra==
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| The [[Dirac matrices]] are a set of four 4×4 [[Matrix (mathematics)|matrices]] that are used as [[Spin (physics)|spin]] and [[Charge (physics)|charge]] [[Operator (physics)|operators]].
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| ===Conventions===
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| There are several choices of [[Signature (physics)|signature]] and [[Group representation|representation]] that are in common use in the physics literature. The Dirac matrices are typically written as <math>\scriptstyle \gamma^\mu</math> where <math>\scriptstyle \mu</math> runs from 0 to 3. In this notation, 0 corresponds to time, and 1 through 3 correspond to x, y, and z.
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| The + − − − [[Signature (physics)|signature]] is sometimes called the [[West Coast of the United States|west coast]] metric, while the − + + + is the [[East Coast of the United States|east coast]] metric. At this time the + − − − signature is in more common use, and our example will use this signature. To switch from one example to the other, multiply all <math>\scriptstyle\gamma^\mu</math> by <math>\scriptstyle i</math>.
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| After choosing the signature, there are many ways of constructing a representation in the 4×4 matrices, and many are in common use. In order to make this example as general as possible we will not specify a representation until the final step. At that time we will substitute in the [[Chirality (physics)|"chiral"]] or [[Hermann Weyl|"Weyl"]] representation as used in the popular graduate textbook ''An Introduction to Quantum Field Theory'' by [[Michael Peskin|Michael E. Peskin]] and [[Daniel Schroeder|Daniel V. Schroeder]].
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| ===Construction of Dirac spinor with a given spin direction and charge===
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| First we choose a [[spin (physics)|spin]] direction for our electron or positron. As with the example of the Pauli algebra discussed above, the spin direction is defined by a [[unit vector]] in 3 dimensions, (a, b, c). Following the convention of Peskin & Schroeder, the spin operator for spin in the (a, b, c) direction is defined as the dot product of (a, b, c) with the vector
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| :<math>(i\gamma^2\gamma^3,\;\;i\gamma^3\gamma^1,\;\;i\gamma^1\gamma^2) = -(\gamma^1,\;\gamma^2,\;\gamma^3)i\gamma^1\gamma^2\gamma^3</math> | |
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| :<math>\sigma_{(a,b,c)} = ia\gamma^2\gamma^3 + ib\gamma^3\gamma^1 + ic\gamma^1\gamma^2</math> | |
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| Note that the above is a [[root of unity]], that is, it squares to 1. Consequently, we can make a [[projection operator]] from it that projects out the sub-algebra of the Dirac algebra that has spin oriented in the (a, b, c) direction:
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| :<math>P_{(a,b,c)} = \frac{1}{2}\left(1 + \sigma_{(a,b,c)}\right)</math>
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| Now we must choose a charge, +1 (positron) or −1 (electron). Following the conventions of Peskin & Schroeder, the operator for charge is <math>\scriptstyle Q \,=\, -\gamma^0</math>, that is, electron states will take an eigenvalue of −1 with respect to this operator while positron states will take an eigenvalue of +1.
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| Note that <math>\scriptstyle Q</math> is also a square root of unity. Furthermore, <math>\scriptstyle Q</math> commutes with <math>\scriptstyle\sigma_{(a, b, c)}</math>. They form a [[complete set of commuting operators]] for the Dirac algebra. Continuing with our example, we look for a representation of an electron with spin in the (a, b, c) direction. Turning <math>\scriptstyle Q</math> into a projection operator for charge = −1, we have
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| :<math>P_{-Q} = \frac{1}{2}\left(1 - Q\right) = \frac{1}{2}\left(1 + \gamma^0\right)</math> | |
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| The projection operator for the spinor we seek is therefore the product of the two projection operators we've found:
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| :<math>P_{(a, b, c)}\;P_{-Q}</math>
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| The above projection operator, when applied to any spinor, will give that part of the spinor that corresponds to the electron state we seek. So we can apply it to a spinor with the value 1 in one of its components, and 0 in the others, which gives a column of the matrix. Continuing the example, we put (a, b, c) = (0, 0, 1) and have
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| :<math>P_{(0, 0, 1)} = \frac{1}{2}\left(1+ i\gamma_1\gamma_2\right)</math>
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| and so our desired projection operator is
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| :<math>P = \frac{1}{2}\left(1+ i\gamma^1\gamma^2\right) \cdot \frac{1}{2}\left(1 + \gamma^0\right) =
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| \frac{1}{4}\left(1+\gamma^0 +i\gamma^1\gamma^2 + i\gamma^0\gamma^1\gamma^2\right)</math>
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| The 4×4 gamma matrices used in the Weyl representation are
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| :<math>\gamma_0 = \begin{bmatrix}0&1\\1&0\end{bmatrix}</math>
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| :<math>\gamma_k = \begin{bmatrix}0&\sigma^k\\ -\sigma^k& 0\end{bmatrix}</math>
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| for k = 1, 2, 3 and where <math>\sigma^i</math> are the usual 2×2 [[Pauli matrices]]. Substituting these in for P gives
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| :<math>P = \frac14\begin{bmatrix}1+\sigma^3&1+\sigma^3\\1+\sigma^3&1+\sigma^3\end{bmatrix}
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| =\frac12\begin{bmatrix}1&0&1&0\\0&0&0&0\\ 1&0&1&0\\0&0&0&0\end{bmatrix}</math> | |
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| Our answer is any non-zero column of the above matrix. The division by two is just a normalization. The first and third columns give the same result:
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| :<math>\left|e^-,\, +\frac12\right\rangle =
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| \begin{bmatrix}1\\0\\1\\0\end{bmatrix}</math>
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| More generally, for electrons and positrons with spin oriented in the (a, b, c) direction, the projection operator is
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| :<math>\frac14\begin{bmatrix}
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| 1+c&a-ib&\pm (1+c)&\pm(a-ib)\\
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| a+ib&1-c&\pm(a+ib)&\pm (1-c)\\
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| \pm (1+c)&\pm(a-ib)&1+c&a-ib\\
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| \pm(a+ib)&\pm (1-c)&a+ib&1-c
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| \end{bmatrix}</math>
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| where the upper signs are for the electron and the lower signs are for the positron. The corresponding spinor can be taken as any non zero column. Since <math>\scriptstyle a^2+b^2+c^2 \,=\, 1</math> the different columns are multiples of the same spinor. The representation of the resulting spinor in the [[Gamma matrices#Dirac basis|Dirac basis]] can be obtained using the rule given in the [[bispinor]] article.
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| ==See also==
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| *[[Dirac equation]]
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| *[[Helicity Basis]]
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| *[[Spin(3,1)]], the [[double cover]] of [[SO(3,1)]] by a [[spin group]]
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| ==References==
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| *{{cite book
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| | last = Aitchison
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| | first = I.J.R.
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| | authorlink =
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| | coauthors = A.J.G. Hey
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| | title = Gauge Theories in Particle Physics (3rd ed.)
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| | publisher = Institute of Physics Publishing
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| |date=September 2002
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| | location =
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| | pages =
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| | url =
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| | doi =
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| | isbn = 0-7503-0864-8 }}
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| * {{Cite web
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| | first = David
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| | last = Miller
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| | title = Relativistic Quantum Mechanics (RQM)
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| | year = 2008
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| | pages = 26–37
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| | url = http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf
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| | postscript = <!--None-->
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| }}
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum field theory]]
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| [[Category:Spinors]]
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| [[Category:Paul Dirac|Spinor]]
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