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# Dirac spinor

dirac spin or, dirac spinors
In quantum field theory, the Dirac spinor is the bispinor in the plane-wave solution

ψ = ω p → e − i p x }\;e^\;}

of the free Dirac equation,

i γ μ ∂ μ − m ψ = 0 , \partial _-m\right\psi =0\;,}

where in the units c = ℏ = 1

ψ is a relativistic spin-1/2 field, ω p → }} is the Dirac spinor related to a plane-wave with wave-vector p → }} , p x ≡ p μ x μ ≡ E t − p → ⋅ x → x^\;\equiv \;Et-}\cdot }} , p μ = \;=\;\left\+}^}},\,}\right\}} is the four-wave-vector of the plane wave, where p → }} is arbitrary, x μ } are the four-coordinates in a given inertial frame of reference

The Dirac spinor for the positive-frequency solution can be written as

ω p → = [ ϕ σ → ⋅ p → E p → + m ϕ ] , }=\phi \\}\cdot }}}+m}}\phi \end}\;,}

where

ϕ is an arbitrary two-spinor, σ → }} are the Pauli matrices, E p → }} is the positive square root E p → = + m 2 + p → 2 }\;=\;++}^}}}

## Contents

• 1 Derivation from Dirac equation
• 11 Results
• 2 Details
• 21 Two-spinors
• 22 Pauli matrices
• 3 Four-spinors
• 31 For particles
• 32 For anti-particles
• 4 Completeness relations
• 5 Dirac spinors and the Dirac algebra
• 51 Conventions
• 52 Construction of Dirac spinor with a given spin direction and charge
• 7 References

## Derivation from Dirac equation

The Dirac equation has the form

− i α → ⋅ ∇ → + β m ψ = i ∂ ψ ∂ t }\cdot }+\beta m\right\psi =i}\,}

In order to derive the form of the four-spinor ω we have to first note the value of the matrices α and β:

α → = [ 0 σ → σ → 0 ] β = [ I 0 0 − I ] }=\mathbf &}\\}&\mathbf \end}\quad \quad \beta =\mathbf &\mathbf \\\mathbf &-\mathbf \end}}

These two 4×4 matrices are related to the Dirac gamma matrices Note that 0 and I are 2×2 matrices here

The next step is to look for solutions of the form

ψ = ω e − i p ⋅ x } ,

while at the same time splitting ω into two two-spinors:

ω = [ ϕ χ ] \phi \\\chi \end}\,}

### Results

Using all of the above information to plug into the Dirac equation results in

E [ ϕ χ ] = [ m I σ → ⋅ p → σ → ⋅ p → − m I ] [ ϕ χ ] \phi \\\chi \end}=m\mathbf &}\cdot }\\}\cdot }&-m\mathbf \end}\phi \\\chi \end}}

This matrix equation is really two coupled equations:

E − m ϕ = σ → ⋅ p → χ E + m χ = σ → ⋅ p → ϕ \leftE-m\right\phi &=\left}\cdot }\right\chi \\\leftE+m\right\chi &=\left}\cdot }\right\phi \end}}

Solve the 2nd equation for χ and one obtains

ω = [ ϕ σ → ⋅ p → E + m ϕ ] \phi \\}\cdot }}}\phi \end}\,}

Alternatively, solve the 1st equation for ϕ and one finds

ω = [ − σ → ⋅ p → − E + m χ χ ] -}\cdot }}}\chi \\\chi \end}\,}

This solution is useful for showing the relation between anti-particle and particle

## Details

### Two-spinors

The most convenient definitions for the two-spinors are:

ϕ 1 = [ 1 0 ] ϕ 2 = [ 0 1 ] =1\\0\end}\quad \quad \phi ^=0\\1\end}\,}

and

χ 1 = [ 0 1 ] χ 2 = [ 1 0 ] =0\\1\end}\quad \quad \chi ^=1\\0\end}\,}

### Pauli matrices

The Pauli matrices are

σ 1 = [ 0 1 1 0 ] σ 2 = [ 0 − i i 0 ] σ 3 = [ 1 0 0 − 1 ] =0&1\\1&0\end}\quad \quad \sigma _=0&-i\\i&0\end}\quad \quad \sigma _=1&0\\0&-1\end}}

Using these, one can calculate:

σ → ⋅ p → = σ 1 p 1 + σ 2 p 2 + σ 3 p 3 = [ p 3 p 1 − i p 2 p 1 + i p 2 − p 3 ] }\cdot }=\sigma _p_+\sigma _p_+\sigma _p_=p_&p_-ip_\\p_+ip_&-p_\end}}

## Four-spinors

### For particles

Particles are defined as having positive energy The normalization for the four-spinor ω is chosen so that the total probability is invariant under Lorentz transformation The total probability is:

P = ∫ V ω † ω d V \omega ^\omega dV}

where V is the volume of integration Under Lorentz transformation, the volume scales as the inverse of Lorentz factor: E / m − 1 } This implies that the probability density must be normalized proportional to E so the total probability is Lorentz invariant The usual convention is to choose ω † ω = 2 E \omega \;=\;2E\,} Hence the spinors, denoted as u are:

u p → , s = E + m [ ϕ s σ → ⋅ p → E + m ϕ s ] },s\right=}\phi ^\\}\cdot }}}\phi ^\end}\,}

where s = 1 or 2 spin "up" or "down"

Explicitly,

u p → , 1 = E + m [ 1 0 p 3 E + m p 1 + i p 2 E + m ] a n d u p → , 2 = E + m [ 0 1 p 1 − i p 2 E + m − p 3 E + m ] },1\right=}1\\0\\}}\\+ip_}}\end}\quad \mathrm \quad u\left},2\right=}0\\1\\-ip_}}\\}}\end}}

### For anti-particles

Anti-particles having positive energy E are defined as particles having negative energy and propagating backward in time Hence changing the sign of E and p → }} in the four-spinor for particles will give the four-spinor for anti-particles:

v p → , s = E + m [ σ → ⋅ p → E + m χ s χ s ] },s=}}\cdot }}}\chi ^\\\chi ^\end}}

Here we choose the χ solutions Explicitly,

v p → , 1 = E + m [ p 1 − i p 2 E + m − p 3 E + m 0 1 ] a n d v p → , 2 = E + m [ p 3 E + m p 1 + i p 2 E + m 1 0 ] },1=}-ip_}}\\}}\\0\\1\end}\quad \mathrm \quad v\left},2\right=}}}\\+ip_}}\\1\\0\\\end}}

Note that these solutions are readily obtained by substituting the ansatz ψ = v e + i p x } into the Dirac equation

## Completeness relations

The completeness relations for the four-spinors u and v are

∑ s = 1 , 2 u p s u ¯ p s = p / + m ∑ s = 1 , 2 v p s v ¯ p s = p / − m \sum _^}_^}&=+m\\\sum _^}_^}&=-m\end}}

where

p / = γ μ p μ =\gamma ^p_\,}      see Feynman slash notation u ¯ = u † γ 0 }=u^\gamma ^\,}

## Dirac spinors and the Dirac algebra

The Dirac matrices are a set of four 4×4 matrices that are used as spin and charge operators

### Conventions

There are several choices of signature and representation that are in common use in the physics literature The Dirac matrices are typically written as γ μ } where μ runs from 0 to 3 In this notation, 0 corresponds to time, and 1 through 3 correspond to x, y, and z

The + − − − signature is sometimes called the west coast metric, while the − + + + is the 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 γ μ } by i

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 "chiral" or "Weyl" representation

### Construction of Dirac spinor with a given spin direction and charge

First we choose a 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

i γ 2 γ 3 , i γ 3 γ 1 , i γ 1 γ 2 = − γ 1 , γ 2 , γ 3 i γ 1 γ 2 γ 3 σ a , b , c = i a γ 2 γ 3 + i b γ 3 γ 1 + i c γ 1 γ 2 i\gamma ^\gamma ^,\;\;i\gamma ^\gamma ^,\;\;i\gamma ^\gamma ^&=-\gamma ^,\;\gamma ^,\;\gamma ^i\gamma ^\gamma ^\gamma ^\\\sigma _&=ia\gamma ^\gamma ^+ib\gamma ^\gamma ^+ic\gamma ^\gamma ^\end}}

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:

P a , b , c = 1 2 1 + σ a , b , c =}\left1+\sigma _\right}

Now we must choose a charge, +1 positron or −1 electron Following the conventions of Peskin & Schroeder, the operator for charge is Q = − γ 0 } , that is, electron states will take an eigenvalue of −1 with respect to this operator while positron states will take an eigenvalue of +1

Note that Q is also a square root of unity Furthermore, Q commutes with σ a , b , c } 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 Q into a projection operator for charge = −1, we have

P − Q = 1 2 1 − Q = 1 2 1 + γ 0 =}\left1-Q\right=}\left1+\gamma ^\right}

The projection operator for the spinor we seek is therefore the product of the two projection operators we've found:

P a , b , c P − Q \;P_}

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

P 0 , 0 , 1 = 1 2 1 + i γ 1 γ 2 =}\left1+i\gamma _\gamma _\right}

and so our desired projection operator is

P = 1 2 1 + i γ 1 γ 2 ⋅ 1 2 1 + γ 0 = 1 4 1 + γ 0 + i γ 1 γ 2 + i γ 0 γ 1 γ 2 }\left1+i\gamma ^\gamma ^\right\cdot }\left1+\gamma ^\right=}\left1+\gamma ^+i\gamma ^\gamma ^+i\gamma ^\gamma ^\gamma ^\right}

The 4×4 gamma matrices used in the Weyl representation are

γ 0 = [ 0 1 1 0 ] γ k = [ 0 σ k − σ k 0 ] \gamma _&=0&1\\1&0\end}\\\gamma _&=0&\sigma ^\\-\sigma ^&0\end}\end}}

for k = 1, 2, 3 and where σ i } are the usual 2×2 Pauli matrices Substituting these in for P gives

P = 1 4 [ 1 + σ 3 1 + σ 3 1 + σ 3 1 + σ 3 ] = 1 2 [ 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ] }1+\sigma ^&1+\sigma ^\\1+\sigma ^&1+\sigma ^\end}=}1&0&1&0\\0&0&0&0\\1&0&1&0\\0&0&0&0\end}}

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:

| e − , + 1 2 ⟩ = [ 1 0 1 0 ] ,\,+}\right\rangle =1\\0\\1\\0\end}}

More generally, for electrons and positrons with spin oriented in the a, b, c direction, the projection operator is

1 4 [ 1 + c a − i b ± 1 + c ± a − i b a + i b 1 − c ± a + i b ± 1 − c ± 1 + c ± a − i b 1 + c a − i b ± a + i b ± 1 − c a + i b 1 − c ] }1+c&a-ib&\pm 1+c&\pm a-ib\\a+ib&1-c&\pm a+ib&\pm 1-c\\\pm 1+c&\pm a-ib&1+c&a-ib\\\pm a+ib&\pm 1-c&a+ib&1-c\end}}

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 a 2 + b 2 + c 2 = 1 +b^+c^\,=\,1} the different columns are multiples of the same spinor The representation of the resulting spinor in the Dirac basis can be obtained using the rule given in the bispinor article

• Dirac equation
• Helicity basis
• Spin3,1, the double cover of SO3,1 by a spin group

## References

• Aitchison, IJR; AJG Hey September 2002 Gauge Theories in Particle Physics 3rd ed Institute of Physics Publishing ISBN 0-7503-0864-8
• Miller, David 2008 "Relativistic Quantum Mechanics RQM" PDF pp 26–37

dirac spin or, dirac spinors

• Dirac spinor beatiful post thanks!

29.10.2014

Dirac spinor
Dirac spinor
Dirac spinor viewing the topic.

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