Download The Dirac Field - SCIPP - University of California, Santa Cruz

Anticommutation Relations and the Exclusion Principle
The Dirac Field
Physics 217 2013, Quantum Field Theory
Michael Dine
Department of Physics
University of California, Santa Cruz
October 2013
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Lorentz Transformation Properties of the Dirac
Field
First, rotations. In ordinary quantum mechanics,
ψ†σi ψ
(1)
is a vector under rotations. How does this work?
Under infinitesimal rotations,
ψ → (1 + iω i S i )ψ
(2)
ψ † σ i ψ → ψ † σ i ψ + iω j ψ † [σ i , σ j ]ψ
(3)
So
= (δ ik − ijk ω j )ψ † σ k ψ
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
This is the transformation law for a vector. We seek an
analogous construction with γ µ .
First we rewrite the transformation law under rotations in a way
that is closer to the form in which we have written infinitesimal
Lorentz transformations. Replace ω i by
ωij = ijk ω k
~ as an
(compare this with rewriting the magnetic field, B,
antisymmetric tensor; also useful in considering Lorentz
transformation properties, Fij ).
Similarly, Jij = ijk J k for the angular momentum operators
(generators of rotations).
Physics 217 2013, Quantum Field Theory
The Dirac Field
(4)
Anticommutation Relations and the Exclusion Principle
In terms of such tensors, the ordinary orbital angular
momentum operator is simply
Lij = −i(xi ∂j − xj ∂i )
(5)
and similarly for other angular momentum operators. The
angular momentum commutation relations are:
[Jij , Jkl ] = i(δik Jjl − δil Jjk − δjk Jil + δjl Jik )
(6)
One can check these commutation relations for the Lij ’s, for
example. One can think of these as the defining relations of the
rotation group. E.g.
eiαij Jij ≈ (1 + iαij Lij + iαij Sij ).
Physics 217 2013, Quantum Field Theory
The Dirac Field
(7)
Anticommutation Relations and the Exclusion Principle
Group Theory of the Lorentz Group
Similarly, we can start with the transformation law for a scalar
under Lorentz transformations:
φ0 (x) = φ(Λ−1 x)
(8)
where Λµν ≈ 1 + ωµν . In other words:
δφ(x) = −ω µν xµ ∂ν φ(x)
(9)
Lµν = i(x µ ∂ ν − x ν ∂ µ )
(10)
So we can define:
as the analog of the generators of orbital rotations; indeed, Lij
are just the angular momentum operators. We can evaluate
their commutators, and abstract the basic commutation
relations for the generators of Lorentz transformations:
[J µν , J ρσ ] − i(g νρ J µσ − g µρ J νσ − g νσ J µρ + g µσ J νρ ).
Physics 217 2013, Quantum Field Theory
The Dirac Field
(11)
Anticommutation Relations and the Exclusion Principle
Representations of the Lorentz Group
We have the analog of orbital angular momentum. For ordinary
rotations, the spin one generator is (G)ijk = −iijk , or in our
antisymmetric index notation:
(Gij )kl = i(δki δlj − δki δ`j )
(12)
The analog for the Lorentz group, corresponding to the
transformation law for vectors, V µ , is
µν
Gαβ
= i(δαµ δβν − δβµ δαν )
(13)
You can check that this:
1
gives the correct infinitesimal transformation for a vector,
xµ
2
obeys the Lorentz group commutation relations.
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Lorentz generators in the spinor representation
Starting with:
{γ µ , γ ν } = 2g µν
(14)
we construct the matrices, analogous to the spin-1/2 matrices:
S µν =
i µ ν
[γ , γ ]
4
(15)
(for µ, ν = i, j it is easy to check that these are the spin
matrices).
These are readily seen to obey the Lie algebra of the Lorentz
group.
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Now, however, it is not ψ † γ µ ψ which transforms as a vector, but
ψ̄γ µ ψ, where
ψ̄ = ψ † γ 0 .
(16)
To check this, we need certain properties of the γ µ matrices,
easily seen to be true in our representations:
1
(γ 0 )† = γ 0 (γ i )† = −γ i
2
From which it follows that: (γ µ )† = γ 0 γ µ γ 0 .
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Let’s start, in fact, by checking that ψ̄ψ is a scalar.
ψ̄ψ → ψ † (1 − iωµν
i
−i 0 ν µ 0 0
γ [γ , γ ]γ )γ (1 + iωρσ [γ ρ , γ σ ])ψ (17)
4
4
= ψ̄ψ.
Now let’s do the vector:
−i 0 ν µ 0 0 µ
i
γ [γ , γ ])γ γ γ (1 + iωρσ [γ ρ , γ σ ])ψ
4
4
(18)
Using the commutation relations:
ψ̄γ µ ψ → ψ † (1 − iωµν
[γ µ , S ρσ ] = Gµρσν γ ν
(19)
the right hand side of eqn. 18 becomes
i
ψ̄γ µ ψ − ωρσ (G ρσ )µν ψ̄γ ν ψ.
2
Physics 217 2013, Quantum Field Theory
The Dirac Field
(20)
Anticommutation Relations and the Exclusion Principle
Introducing also the matrix
γ 5 = iγ 0 γ 1 γ 2 γ 3
(21)
which anti commutes with all of the other γ’s, we can construct
the following bilinears in the fermion field which transform as
irreducible tensors:
1
Scalar: ψ̄ψ
2
Pseudoscalar: ψ̄γ 5 ψ
3
Vector: ψ̄γ µ ψ
4
Pseudo vector: ψ̄γ µ γ 5 ψ
5
Second rank tensor: ψ̄σ µν ψ.
(You will get to familiarize yourselves with these objects for
homework; the "pseudo" character will be discussed shortly.)
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
With these results, it is easy to construct a relatiavitsically
invariant lagrangian:
L = i ψ̄∂µ γ µ ψ − mψ̄ψ ≡ i ψ̄ 6 ∂ ψ − mψ̄ψ.
Euler-Lagrange equations (varying with respect to ψ, ψ̄
independently:
i 6 ∂ ψ − mψ = 0
the Dirac equation.
We want to interpret now as quantum fields.
Physics 217 2013, Quantum Field Theory
The Dirac Field
(22)
(23)
Anticommutation Relations and the Exclusion Principle
The canonical momentum is curious:
Π=
∂L
= iψ †
∂ ψ̇
(24)
With this we can construct the Hamiltonian; it is reminiscent of
Dirac’s original expression:
Z
Z
3
~ + βm)ψ(x).
H = d xH = d 3 xψ † (x)(−iγ 0~γ · ∇
(25)
We will take a step which we will see is necessary for a
sensible interpretation of the theory: we require
{ψ(~x , t), Π(~x 0 , t)} = iδ(~x − ~x 0 ).
(26)
In order to develop a momentum space expansion of the
fermion fields, as we did for scalars, we first, we need more
control of the solutions of the free field equations in momentum
space (for the scalar, these were trivial).
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Momentum Space Spinors
In your text, various relations for spinors, including
orthogonality relations and spin sums, are worked out by
looking at explicit solutions. We can short circuit these
calculations in a variety of ways. Here is one:
Two things slightly different than your text:
1
Dirac matrices: It will be helpful to have an explicit
representation of the Dirac matrices, or more specifically of
Dirac’s matrices, somewhat different than the one in your
text:
1 0
0 ~σ
0
γ =
~γ =
(27)
0 −1
−~σ 0
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Quick calculation of spin sums, Normalizations,
etc.
We first consider the positive energy apinors, p0 =
If χ is a constant spinor,
p
~p2 + m2 .
u(p) = N(6 p + m)χ
solves the Dirac equation. Now take χ to be a solution of the
Dirac equation with ~p = 0. We can work, for this discussion, in
any basis, so let’s choose our original basis, where the ~p = 0
spinors are particularly simple, and take the two
linearly-independent spinors to be
 
 
1
0
0
1

 
χ1 = 
0 ; χ2 = 0
0
0
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Let’s first get the normalizations straight. We will require:
ū(p)u(p) = 2m
(note this is Lorentz invariant). With our solution the left hand
side is
N 2 χ† (6 p† + m)γ o (6 p + m)χ
= N 2 χ† γ o γ o (6 p† + m)γ o (6 p + m)χ
= N 2 χ† γ o (6 p + m)(6 p + m)χ
= N 2 χ† 2m(6 p + m)χ
From the explicit form of the Dirac matrices and the χ’s,
χ† 6 pχ = E. So
1
.
N2 =
(E + m)
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
With this we can do the spin sums. First note that for the χ’s, looking
at their explicit form:
X
1
1 0
χχ† =
= (1 + γ o ).
0 0
2
s
So now
X
u(p, s)ū(p, s) =
s
X
u(p, s)u † (p, s)γ o
s
=
=
1 2
N (6 p + m)(1 + γ o )(6 p† + m)γ o
2
1 2
N (6 p + m)(1 + γ o )γ o γ o (6 p† + m)γ o
2
1
= N 2 (6 p + m)(1 + γ o )(6 p + m)
2
1
= N 2 2(m + po )(6 p + m)
2
= (6 p + m).
(in the next to last step, just multiply out the terms).
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Finally, we can compute the inner products:
u † (p, s)u(p, s0 ) = N 2 χ† (6 p† + m)(6 p + m)χ
= N 2 χ† γ o (6 p + m)γ o (6 p + m)χ
= N 2 χ† γ o (p2 − m2 + 2po (6 p + m)γ o χ
= 2Eδs,s0
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Exercise:
Work out the corresponding relations for the negative energy
spinors, v (p, s), including the spin sums, normalization, and
orthogonality relations.
X
X
uα (p, s)ūβ (p, s) = (6 p+m)αβ
vα (p, s)v̄β (p, s) = (6 p−m)αβ
s
s
uα† (p, s)uα (p, s0 ) = 2Ep δss0 = vα† (p, s)vα (p, s0 )
ūα (p, s)uα (p, s0 ) = 2mδss0 = v̄α (p, s)vα (p, s0 )
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Now we want to write momentum space expansions of the
spinor fields, analogous to those for scalar fields. Suppose we
have operators a, a† which obey the anticommutation relations:
{a, a† } = 1; {a, a} = 0; {a† , a† } = 0
Then construct the “number operator"
N = a† a
and the state |0i by the condition
a|0i = 0
(a is a destruction operator). Then
Na† |0i = a† aa† |0i.
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Using the anticommutation relations to move a to the right,
= −a† a† a|0i + a† |0i.
In other words
Na† |0i = |0i
so a† creates a one particle state. But since (a† )2 = 0 (from the
anticommutation relations) there is no two particle state.
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
So anticommutation relations of this kind build in the exclusion
principle; only the zero and one-particle states are allowed.
Consider, first, finite volume, as we did for the scalar field. The
field operator Ψ(~x , t) should satisfy the Dirac equation. So we
write
XX 1 p
ψα =
a(p, s)uα (p, s)e−ip·x + b† (p, s)vα (p, s)eip·x .
2Ep
s
p
(28)
XX 1 †
†
−ip·x
†
†
p
ψα =
a (p, s)uα (p, s)e
+ b(p, s)vα (p, s)eip·x .
2Ep
s
p
(29)
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Taking
{a(~p, s), a† (~p0 , s0 )} = δss0 δ~p,~p0
satisfies the (anti) commutation relations above.
For the Hamiltonian we obtain (as we will see in a moment),
X
H=
E(p)(a† (p, s)a(p, s) + b† (p, s)b(p, s)) + ∞
~p,s
only because we assumed anti commutation relations. So we
see that we can have states with zero or one electron and zero
or one positron for each momentum and spin.
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Momentum Space Expansion of the Dirac Field,
Infinite volume
Infinite volume:
ψα =
XZ
s
d 3p
√
(a(p, s)uα (p, s)e−ip·x
(2π)3 2Ep
(30)
+b† (p, s)vα (p, s)eip·x ).
ψα†
=
XZ
s
d 3p
√
(a† (p, s)uα† (p, s)e−ip·x
(2π)3 2Ep
+b(p, s)vα† (p, s)eip·x ).
Physics 217 2013, Quantum Field Theory
The Dirac Field
(31)
Anticommutation Relations and the Exclusion Principle
Now consider the Hamiltonian:
Z
~ + m ψ.
H = d 3 x ψ̄ −i~γ · ∇
(32)
Plug in the expansions of the fields, and use:
(~γ · ~p + m)u = p0 γ 0 u
(−~γ · ~p + m)v = −p0 γ 0 v
(33)
and
u † (p, s)u(p, s0 ) = v † (p, s)v (p, s0 ) = 2p0 δs,s0
(34)
i
d 3p X h †
†
E
a
(p,
s)a(p,
s)
−
b(p,
s)b
(p,
s)
.
p
(2π)3 s
(35)
to find:
Z
H=
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Now if we quantize as for the scalar field with commutators, we
obtain a negative contribution from the negative energy states,
and the energy is unbounded below. If we quantize with anti
commutators, we have:
Z
i
d 3p X h †
†
H=
E
a
(p,
s)a(p,
s)
+
b
(p,
s)b(p,
s)
−
1
.
p
(2π)3 s
(36)
So we now have a sensible expression in terms of number
operators, with an infinite contribution which we can think of as
representing the energy of the Dirac sea.
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
The Dirac Propagator
Just as Dirac fields obey anti commutation relations, the
time-ordered product for fermions is designed with an extra
minus sign, for example:
T (ψ(x1 )ψ(x2 )) = θ(x10 − x20 )ψ(x1 )ψ(x2 ) − θ(x20 − x10 )ψ(x2 )ψ(x1 ).
(37)
The basic fermion propagator is:
SF (x1 − x2 ) = T h0|ψ(x)ψ̄(y )|0i.
Physics 217 2013, Quantum Field Theory
The Dirac Field
(38)
Anticommutation Relations and the Exclusion Principle
Let’s take a particular time ordering, x10 > x20 ,
d 3 pd 3 p0
p
e−ip·(x1 −x2 ) u(p, s)ū(p0 , s0 ).
6 E(p)E(p 0 )
(2π)
s,s0
(39)
The x integral gives δ(p − p0 ); then we have, from the sum over
spins, 6 p + m. Indeed, up to the factor of 6 p + m, this is what we
had for the same time ordering for the scalar propagator.
SF =
XZ
d 4x
For the other time ordering, we obtain, again, the same result
as for the scalar, except with a factor − 6 p + m, from the spin
sum. Changing p → −p gives
SF (p) =
Physics 217 2013, Quantum Field Theory
p2
6p + m
.
− m2 + i
The Dirac Field
(40)
Anticommutation Relations and the Exclusion Principle
The Discrete Symmetries P and C
Parity: The Dirac lagrangian is unchanged if we make the
replacement:
ψ(~x , t) → γ o ψ(−~x , t)
(41)
Let’s see what effect this has on the creation and annihilation
operators, a, b, etc.
ψp (~x , t) = γ o ψ(−~x , t)
(42)
Z
o o
o o
d 3p
p (a(~p, s)γ o u(~p, s)e−ip x −i~p·~x +b† (~p, s)γ o v (~p, s)eip x +i~p·~x ).
=
(2π)3 Ep
We can easily determine what γ o does to u and v using our explicit
expressions (ignoring the normalization factor, which is unimportant
for this discussion):
γ o (6 p + m)χ = (po γ o + ~p · ~γ + m)γ o χ
(43)
= u(−~p, s)
o
γ v (~p, s) = −v (−~p, s)
Physics 217 2013, Quantum Field Theory
The Dirac Field
(44)
Anticommutation Relations and the Exclusion Principle
So making the change of variables ~p → −~p in our expression
for ψp , gives
ψp (~x , t) = γ o ψ(−~x , t)
(45)
Z
d 3p
p (a(−~p, s)u(~p, s)e−ip·x + b† (−~p, s)v (~p, s)eip·x )
=
3
(2π) Ep
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Charge Conjugation: Now we can do the same thing for C.
Here:
ψc (x) = γ 2 ψ ∗ (x)
(46)
Z
d 3p
p (a† (~p, s)γ 2 u ∗ (~p, s)eip·x +b(~p, s)γ 2 v ∗ (~p, s)e−ip·x ).
=
3
(2π) Ep
Now we consider the action of γ 2 on u ∗ , v ∗ :
γ 2 u ∗ = γ 2 (6 p∗ +m)χ∗ = γ 2 (po γ o −p1 γ 1 +p2 γ 2 −p3 γ 3 +m)χ (47)
= (− 6 p + m)γ 2 χ.
Physics 217 2013, Quantum Field Theory
The Dirac Field
Anticommutation Relations and the Exclusion Principle
Here we have not been ashamed to use the explicit properties
of the γ matrices; γ 1 and γ 3 are real, while γ 2 is imaginary; the
first two anticommute with γ 2 while the third commutes. Now
we use the explicit form of γ 2 to see that it takes the positive
energy χ to the negative energy χ, with the opposite spin. So,
indeed, we have that
ac (p, s) = b(p, −s)
bc (p, s) = a(p, −s)
i.e. it reverses particles and antiparticles and flips the spin.
Exercise: Determine the action of P and C on particle and
antiparitcle states of definite momentum.
Physics 217 2013, Quantum Field Theory
The Dirac Field
(48)