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Units, Constants and Useful Formulas
Relativistic Quantum Field Theory
---------------------------------
Canonical Representation
------------------------
One of the key insights of quantum field theory is to do away with
classical mathematical forms of the wave function by listing how
many particles (quanta) are in each momentum state, k. These
occupation numbers are the analog of the Slater determinant in QM.
This referred to as the CANONICAL REPRESENTATION of QFT.
We can specify the state of a system as:
|n(p_{1}) n(p_{2}) n(p_{2}) .... n(p_{∞})>
where,
n(p_{i}) = number of quanta with momentum p_{i}
These states are referred to as FOCK States.
Examples:
|0> = |0,0,0,0,0>
a_{1}^{†}|0> = |1,0,0,0,0>
a_{2}^{†}a_{1}|0> = |0,1,0,0,0>
|0> = |0,0,0,0,0>
a_{1}^{†}a_{1}^{†}|0> = |2,0,0,0,0>
Consider the periodic loop. Momentum (and frequency) are quantized where
p = 2πn/L. The waves moving on the loop are equivalent to a collection
of harmonic oscillators - one harmonic oscillator for each value of n.
For each harmonic there is a number of quanta n(p_{n}) represented by the
occupation number.
Field Operators
---------------
In QM, the general solution to the wave equation is a superposition
of plane waves (Fourier expansion of basis vectors):
ψ(x) = Σ_{p}A_{p}exp(ipx) where A_{p} is an amplitude term.
and,
ψ^{*}(x) = Σ_{p}A_{p}^{*}exp(-ipx)
By analogy, in QFT we can write:
Ψ^{†}(x) = Σ_{p}a_{p}^{†}exp(-ipx) where a_{p}^{†} is the creation operator
and
Ψ(x) = Σ_{p}a_{p}exp(ipx) where a_{p} is the annihilation operator
The field operator is not the same thing as a single-particle
wavefunction. The former is an operator acting on the Fock
space, and the latter is a quantum-mechanical amplitude for
finding a particle in some position. However, they are closely
related. We use capitals to denotes the field operators.
Ψ^{†}(x)/Ψ(x) create/destroy a particle at a particular point in
space. a_{p}^{†}/a_{p} create/annihilate particles with certain momenta.
Field operators act by applying the Fourier transform to the
creation and annihilation operators.
Examples:
Pre-existing particle, n_{5} = 1
|0 0 0 0 1 0 0>
Create a new particle in momentum state p = 3 at x. Thus,
n_{3} -> 1, n_{5} = 1
Σ_{p}a_{3}^{†}exp(-i3x)|0 0 0 0 1 0 0> => e^{-i3x}|0 0 1 0 1 0 0>
Create another new particle in momentum state p = 5 at the same x.
Thus, n_{5} -> 2
Σ_{p}a_{5}^{†}exp(-i5x)|0 0 1 0 1 0 0> => √2e^{-i5x}|0 0 1 0 2 0 0>
The factor of √2 comes from a^{†}|n> = √(n+1)|n+1>
Creation and Annihilation Operators on Bra Vectors
--------------------------------------------------
The rule for the creation and annihilation operators when they
operate on bra vectors is the opposite of the case for ket
vectors. Thus, the creation operator acts on a bra as an
annihilation operator and vice versa.
<n|a^{†} => <n-1|√n
and
<n|a => <n+1|√n+1
What is the meaning of ∫Ψ^{†}(x)Ψ(x)dx ?
-------------------------------------
Expand in terms of basis states.
= ∫Σ_{p}a_{p}^{†}e^{-ipx}Σ_{q}a_{q}e^{iqx}dx
= ∫Σ_{p}a_{p}^{†}Σ_{q}a_{q}e^{i(q-p)x}dx
= ∫Σ_{p}a_{p}^{†}Σ_{q}a_{q}δ_{pq}dx
= ∫Σ_{p}a_{p}^{†}a_{p} - the occupation number is a^{†}_{p}a_{p}
= the number of particles at x.
Therefore, Ψ^{†}(x)Ψ(x) is the density of particles at x.
Time Dependent Schrodinger Equation
-----------------------------------
Differentiate, Ψ w.r.t. t and x:
∂Ψ/∂t = Σ_{p}(-iω)a_{p}exp(i(px - ωt))
∂Ψ/∂x = Σ_{p}(ip)a_{p}exp(i(px - ωt))
∂^{2}Ψ/∂x^{2} = Σ_{p}(ip)^{2}a_{p}exp(i(px - ωt))
So we can write:
∂Ψ/∂t/∂^{2}Ψ/∂x^{2} = Σ_{p}(-iω)a_{p}exp(i(px - ωt))/Σ_{p}(ip)^{2}a_{p}exp(i(px - ωt))
^{ }^{ }= (-iω)/(ip)^{2}
Which leads to:
^{ }^{ }(ip)^{2}∂Ψ/∂t = (-iω)∂^{2}Ψ/∂x^{2}
^{ }^{ }p^{2}∂Ψ/∂t = (iω)∂^{2}Ψ/∂x^{2}
Now E = hω = p^{2}/2m
Therefore p^{2} = 2mω/h
(2mω/h)∂Ψ/∂t/∂ = (iω)∂^{2}Ψ/∂x^{2}
∂Ψ/∂t =(ih/2m)∂^{2}Ψ/∂x^{2}
Note: In both cases we could have equally integrated over
position/time to demonstrate conservation of momentum and
conservation of energy respectively.
Heisenberg Uncertainty Principle in QFT
---------------------------------------
Analagous to the relationship in QM, the creation and annihilation
for position and momentum are Fourier transforms of each other.
Thus:
Ψ(x)^{†} = Σ_{p}a_{p}^{†}exp(-ipx) and Ψ(x) = Σ_{p}a_{p}exp(ipx)
and
a_{p}^{†} = Σ_{x}Ψ(x)^{†}exp(ipx) and a_{k} = Σ_{x}Ψ(x)exp(-ipx)
This shows that to create a state of definite position, it is
necessary to sum over many momentum states. Conversely, to create
a state of definite momentum it is necessary to sum over many
position states.
Free Field Quantization
-----------------------
It is well known that the standard equations in field theory for
scalar and fermionic fields are respectively the Klein-Gordon and
Dirac equations.
Massive Spin 0 Real Scalar Fields
---------------------------------
Consider the Klein-Gordon equation (h = c = 1):
∂^{2}φ/∂t^{2} - ∇^{2}φ + m^{2}φ = 0
This can be derived using the relativistic relationship (h = c = 1):
E^{2} - p^{2} - m^{2} = 0 <=> -η^{μν}p_{μ}p_{ν} + m^{2} = 0 where η^{μν} = (-, +, +, +)
and
p_{μ} -> i∂/∂x^{μ} (p_{0} -> i∂/∂t)
In QM solutions to the K-G equation are of the form φ = exp(i(kx - ωt))
with the constraint that:
ω = √(p^{2} + m^{2}) {from E = hω = √(p^{2}c^{2} + m^{2}c^{4})}
Canonical Commutation Relations
-------------------------------
Let's look at the commutator, [x,p] = ihδ_{ij}. In QM this is referred
to as the CANONICAL COMMUTATION RELATION and defines the relationship
between canonical conjugate quantities (quantities which are related
through the Fourier transform). The quantity p is often written as π
and is referred to as the momentum conjugate to x. By analogy with
this we can write a similar relationship for the commutators for
quantum fields:
[φ(x),π(y)] = ihδ^{(3)}(x - y)
[φ(x),φ(y)] = [π(x),π(y)] = 0
The Lagrangian for the K-G equation is (h = c = 1):
L = (1/2)∂_{μ}φ∂^{μ}φ - (1/2)m^{2}φ^{2}
The quantity, π, for the fields is obtained using the Euler-Lagrange
equations generally written as:
∂_{μ}(∂L/∂(∂_{μ}φ)) - ∂L/∂φ = 0
The momenta canonically conjugate to the coordinates of the
field (or simply the canonical momentum) is defined as:
.
π = ∂L/∂φ
For the K-G equation we get:
. .
π = ∂L/∂φ = φ
Therefore:
.
[φ(x),π(y)] ≡ [φ(x),φ(y)] = ihδ^{3}(x - y)
The quantization of the harmonic oscillator can now be applied to the
free scalar field by writing φ as a linear sum (Fourier expansion) of
creation and annihilation operators. Using the expessions for the
creation and annihilation operators for the harmonic oscillator, we
can write:
a_{p}^{†} = √(ω/2)(φ + iπ/√(2ω) or (1/√(2ω)(ωφ + iπ)
and
a_{p} = √(ω/2)(φ - iπ/√(2ω) or (1/√(2ω)(ωφ - iπ)
Therefore,
φ = (1/√(2ω))(a_{p}exp(i(px - ωt) + a_{p}^{†}exp(-i(px - ωt))
and
.
π = φ = -i√(ω/2)(a_{p}exp(i(px - ωt) - a_{p}^{†}exp(-i(px - ωt))
Thus, the field operator equations are:
φ(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))[a_{p}exp(ipx) + a_{p}^{†}exp(-ipx)]
and
.
π(x) = φ(x) = ∫(d^{3}p/(2π)^{3}) (-i√(ω/2))[a_{p}exp(ipx) - a_{p}^{†}exp(-ipx)]
The commutator [a_{p},a_{q}^{†}] is a little tricky to derive. It can be
shown to be:
[a_{p},a_{q}^{†}] = (2π)^{3}δ^{(3)}(p - q)
These creation and annihilation operators represent a more abstract
form of the operators that we derived for the Quantum Harmonic
oscillator.
We can summarize the commutators for the real scalar field as
follows:
[a_{p},a_{q}^{†}] = (2π)^{3}δ^{(3)}(p - q)
[φ(x),π(y)] = ihδ^{(3)}(x - y)
[a_{p},a_{q}] = [a_{p}^{†},a_{q}^{†}] = 0
[φ(x),φ(y)] = [π(x),π(y)] = 0
The operators φ(x) and π(y) depend on space and not time. This is
the SCHRODINGER PICTURE. In the HEISENBERG PICTURE the operators
change with time. However, the operators in the two pictures agree
at a fixed time, say, t = 0. The commutation relations become EQUAL
TIME commutation relations in the Heisenberg picture.
[φ(x,t),π(y,t)] = ihδ^{(3)}(x - y)
[φ(x,t),φ(y,t)] = [π(x),π(y)] = 0
We can show this as follows:
a_{p}(t) = exp(iHt)a_{p}(0)exp(-iHt)
Now, from before we had:
H = ωa_{p}^{†}a_{p}
Therefore,
i[H,a_{p}(t)] = -iωa_{p}(t)
[H,a_{p}(t)] = -ωa_{p}(t)
Similarly,
[H,a_{p}^{†}(t)] = ωa_{p}^{†}(t)
Plugging these into the Heisenberg equation, i[H,a_{p}(t)] = ∂a_{p}(t)/∂t,
yields:
∂a_{p}(t)/∂t = -ωa_{p}(0)
with solution,
a_{p}(t) = a_{p}(0)exp(-iωt)
and,
∂a_{p}^{†}(t)/∂t = ωa_{p}^{†}(t)
with solution,
a_{p}^{†}(t) = a_{p}^{†}(0)exp(iωt)
Thus, the field operator equations in the Heisenberg picture are:
φ(x,t) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))[exp(-iωt)a_{p}exp(ipx) + exp(iωt)a_{p}^{†}exp(-ipx)]
= ∫(d^{3}p/(2π)^{3}) (1/√(2ω))[a_{p}exp(ip.x) + a_{p}^{†}exp(-ip.x)]
Where p is now a 4-vector instead of a 3-vector.
p.x = px - ωt
It is easy to that this satisfies the K-G equation.
.
Now, we said from before that π = φ. Therefore,
.
φ(x,t) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))[-iωa_{p}exp(ip.x) + iωa_{p}^{†}exp(-ip.x)]
= ∫(d^{3}p/(2π)^{3}) (1/√(2ω))(-iω)[a_{p}exp(ip.x) - a_{p}^{†}exp(-ip.x)]
= ∫(d^{3}p/(2π)^{3}) (-i√(ω/2))[a_{p}exp(ip.x) - a_{p}^{†}exp(-ip.x)]
= π
Which is what we had before, so everything checks out!
Non Relativistic Limit
----------------------
In the non-relativistic limit the K-G equation yields the Schrodinger
equation. We can see this by replacing E = √(p^{2} + m^{2}) with the
non-relativistic relationship E = p^{2}/2m. We get:
ih∂_{t}φ - (1/2m)∇^{2}φ = 0
The Lagrangian for the SE is:
L = φ^{†}(ih∂_{t} - (h^{2}/2m)∇^{2})φ
Where,
.
π = ∂L/∂φ = iφ^{†}
Therefore:
[φ(x),π(y)] ≡ [φ(x),iψ(y)]
Now,
[φ(x),iψ(y)] = φ(x)iφ(y) - iφ(y)φ(x)
_{ } = i(φ(x)φ(y) - φ(y)φ(x))
_{ } = i[φ(x),φ(y)]
Therefore, by comparison:
[φ(x),φ(y)] = hδ(x - y)
Relativistic Normalization
--------------------------
The formulation involving 4-vectors is clearly Lorentz invariant.
But what about 3-vectors?
The Dirac Delta function in 3 dimensions is defined as:
δ^{(3)}(p - q) = (1/2π)^{3}∫ d^{3}x exp(i(p - q).x)
Likewise in spacetime it is defined as:
δ^{(4)}(p - q) = (1/2π)^{4}∫ d^{4}x exp(i(p - q).x)
δ^{(4)} is Lorentz invariant but δ^{(3)} is not. How do we make it so?
Consider:
δ^{(4)}(p - q) = δ(E_{p} - E_{q})δ^{(3)}(p - q)
Now consider;
δ(E_{p}^{2} - E_{q}^{2})
Using the following property of the delta function:
δ(x^{2} - α^{2}) = (1/|2α|)[δ(x + α) + δ(x - α)]
We can write:
δ(E_{p}^{2} - E_{q}^{2}) = (1/|2E_{q}|)[δ(E_{p} + E_{q}) + δ(E_{p} - E_{q})]
If we disregard the negative energies we get:
δ(E_{p} - E_{q}) = 2E_{q}δ(E_{p}^{2} - E_{q}^{2})
Therefore,
δ^{(4)}(p - q) = 2E_{q}δ(E_{p}^{2} - E_{q}^{2})δ^{(3)}(p - q)
Now δ(E_{p}^{2} - E_{q}^{2}) is Lorentz invariant so the implication is that the
invariant form of δ^{(3)}(p - q) is 2E_{q}δ^{(3)}(p - q)
Now by definition ∫d^{3}p δ^{(3)}(p - q) = 1. Therefore, to compensate for
the Lorentz invariant for we need to divide by 2E_{q}. Thus,
∫(d^{3}p/2E_{q}) 2E_{q}δ^{(3)}(p - q) = 1
Therefore, the relativistically normalized momentum states are
given by:
<p|q> = (2π)^{3}2ωδ^{(3)}(p - q)
|p> = √(2ω)|p> = √(2ω)a_{p}^{†}|0>
<q| = <q|√(2ω) = <0|√(2ω)a_{p}
Note: Some texts also define relativistically normalized creation
operators as:
a_{p} = a(p)/√(2ω).
This results in the 1/2ω factor seen in these texts. We will not
assume this throughout this document.
The Scalar Propagator
---------------------
We can use the canonical approach to write the propagator as:
G(x,y) = <0|annihilate particle at x, create particle at y|0>
Thus, y -> x
The propagator can now be written as:
G(x,y) = <0|φ(x)φ(y)|0>
By plugging in the equations for φ(x), φ(y) we end up with:
<0|φ(x)φ(y)|0> = ∫(d^{3}pd^{3}q/(2π)^{6}) (1/√(4ω_{p}ω_{q}))<0|a_{p},a_{a}^{†}|0>exp(-ipx + iqy)
Which, after some math, yields:
<0|φ(x)φ(y)|0> = ∫(d^{3}p/(2π)^{3}) (1/2ω)exp(-ip(x - y))
Until now, we integrated only over 3-momentum, with p_{0} fixed by the
'on-shell' relationship, E^{2} = ω^{2} = p_{0}^{2} = p^{2} + m^{2}. To accomodate 'off-shell'
virtual particles we need to integrate over the 4-momentum. This
also makes the propagator manifestly Lorentz invariant. Therefore,
we now integrate over 4-dimensional Minkowski spacetime. Thus,
G(x,y) = ∫(d^{4}p/(2π)^{4}) exp(-ip(x-y))/(p^{2} - m^{2})
However, as it stands this integral is ill-defined because the
integrand has 2 poles at p^{0} = ±√(p^{2} + m^{2}). The final expression
for the propagator will depend on the choice of contour. A
contour going under the left pole and over the right pole gives
the FEYNMAN PROPAGATOR. Thus, the contour looks like:
Instead of specifying the contour, it is standard to write the
Feynman propagator as:
G(x,y) = ∫(d^{4}p/(2π)^{4}) exp(-ip(x-y))/(p^{2} - m^{2} + iε)
This way of writing the propagator is called the "iε prescription".
The iε term has the effect of shifting the poles slightly off
the real axis, so the integral is equivalent to the contour along
the real p^{0} axis.
The Feynman propagator satisfies:
(∂^{2}φ/∂t^{2} - ∇^{2}φ + m^{2})G(x,y) = -iδ^{(4)}(x - y)
So G(x,y) is a Green's function for the Klein-Gordon operator.
Solutions to G(x,y) over the chosen contour yield:
G_{F}(x,y) = lim_{ε->0}∫(d^{4}p/(2π)^{4}) exp(-ip(x-y)/(p^{2} - m^{2} + iε)
_{ } = -(1/4π)δ(s) + (m/8π√s)H_{1}^{(2)}(m√s) for s ≥ 0
_{ } = -(im/4π^{2}√(-s)K_{1}(m√(-s)) for s < 0
Where s := (x^{0} - y^{0}) - (x - y), H_{1}^{(2)}(m√s) is a Hankel function
and K_{1}(m√(-s)) is a modified Bessel function.
The Fourier transform of the position space propagators can be
thought of as propagators in momentum space. This is simpler
than the position space form.
G_{F}(p) = -i/(p^{2} - m^{2} + iε)
Massive Spin 0 Complex Scalar Fields
------------------------------------
Complex fields are required to accomodate charge. φ carries a charge
of +1. φ^{†} carries a charge of -1
We can write φ in terms of real and imaginary parts.
φ = (1/√2)(φ_{R} + iφ_{I})
The K-G Lagrangian becomes (h = c = 1):
L = ∂_{μ}φ∂^{μ}φ^{†} - m^{2}φφ^{†}
= (1/2)∂_{μ}φ_{R}∂^{μ}φ_{R} + (1/2)∂_{μ}φ_{L}∂^{μ}φ_{L} - (1/2)m^{2}φ_{R}^{2} - (1/2)m^{2}φ_{L}^{2}
Which is 2 uncoupled real scalar fields.
. . . .
π = ∂L/∂φ = φ^{†} and π^{†} = ∂L/∂φ^{†} = φ
Therefore:
.
[φ(x),π(y)] ≡ [φ(x),φ^{†}(y)]
The equation of motion associated to the above Lagrangian is the
very same Klein-Gordon equation. However, now complex solutions
of φ(x) are allowed.
In order to describe boson particles with electric charge we need
to introduce a new pair of creation and annihilation operators, b^{†}
and b which create and annihilate antiparticles. Therefore,
b_{p}^{†} creates a particle of momentum p.
c_{p}^{†} creates an antiparticle of momentum p.
The field operator equations are:
φ(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))[b_{p}exp(ip.x) + c_{p}^{†}exp(-ip.x)]
φ^{†}(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))[b_{p}^{†}exp(-ip.x) + c_{p}exp(ip.x)]
.
π(x) = φ^{†}(x) = ∫(d^{3}p/(2π)^{3}) (i(√(ω/2))[b_{p}^{†}exp(-ip.x) - c_{p}exp(ip.x)]
^{ } .
π^{†}(x) = φ(x) = ∫(d^{3}p/(2π)^{3}) (-i(√(ω/2))[b_{p}exp(ip.x) - c_{p}^{†}exp(-ip.x)]
We can summarize the same-time commutators for the complex field
as follows:
[b_{p},b_{q}^{†}] = [c_{p},c_{q}^{†}] = (2π)^{3}δ^{(3)}(p - q)
[φ(x),π(y)] = [φ^{†}(x),π^{†}(y)] = ihδ(x - y)
All other combinations of b_{p}, b_{p}^{†}, c_{p}, c_{p}^{†} = 0
All other combinations of φ(x), φ^{†}(x), π(y), π(y)^{†} = 0
Conserved Quantities
--------------------
It is easy to see that the Lagrangian is invariant under the
transformation (φ -> φexp(iθ) and φ^{*} -> φ^{*}exp(-iθ)). This
implies there is a symmetry and a conserved quantity.
Now e^{iε} = 1 + iε ... Taylor series
Thus,
φ -> φ + iεφ -> φ + εf_{φ} where f_{φ} = iφ
φ^{*} -> φ^{*} - iεφ^{*} -> φ^{*} - εf_{φ*} where f_{φ*} = -iφ^{*}
Look for a conserved quantity using Noether's Theorem.
. .
π_{φ} = ∂L/∂φ = φ^{*}
. .
π_{φ*} = ∂L/∂φ^{*} = φ
Conserved quantity = ∫[π_{φ}f_{φ} + π_{φ*}f_{φ*}]d^{3}x
. .
= i∫[φ^{*}φ - φφ^{*}]d^{3}x
= i∫[πφ - π^{*}φ^{*}]d^{3}x
The quantity in [] is the charge density of the field, ρ.
Another way to write this is:
.
ρ = Im(φφ^{*})
∴ Q = ∫ρd^{3}x
We can also write a 4-vector for current, j^{μ}, as:
j^{μ} = Im(φ^{*}∂^{μ}φ)
∂_{μ}j^{μ} = Im(∂_{μ}(φ^{*}∂^{μ}φ))
∂_{μ}j^{μ} = Im(∂_{μ}φ^{*}∂^{μ}φ + φ^{*}∂_{μ}∂^{μ}φ)
The first term is real so the imaginary part is 0. The second
term can be written as:
-φ^{*}m^{2}φ since ∂_{μ}∂^{μ}φ + m^{2}φ = 0
Again this is real. We can therefore conclude that:
∂_{μ}j^{μ} = 0
This is the CONTINUITY EQUATION.
Substituting the equations for the field operators yields:
Q = ∫d^{3}p/(2π)^{3} (c_{p}^{†}c_{p} - b_{p}^{†}b_{p})
The Complex Scalar Propagator
-----------------------------
The real scalar propagator describes the creation of a particle
at y and its annihilation at x. The question is how how to deal
with antiparticles^{*}. Feynman decided that the propagator must
be made up of 2 parts - particles are represented by the case
x^{0} > y^{0} and antiparticles are represented by x^{0} < y^{0}. In order
to include the latter it is necessary to introduce the WICK TIME
0RDERING symbol, T. T is defined as:
Tφ(x)φ(y) = φ(x)φ(y) for x^{0} > y^{0}
= φ(y)φ(x) for x^{0} < y^{0}
The propagator can now be written as:
G(x,y) = <0|Tφ(x)φ(y)|0>
= <0|Θ(x^{0} - y^{0})φ(x)φ(y) + Θ(y^{0} - x^{0})φ(y)φ(x)|0>
^{*} Although we need complex fields to represent antiparticles, this
interpretation is also there for a real scalar field because the
particle is its own antiparticle.
For a complex field:
G(x,y) = <0|Tφ(x)φ^{†}(y)|0>
Where,
Tφ(x)φ(y)^{†} = φ(x)φ(y)^{†} for x^{0} > y^{0}
^{ } = φ^{†}(y)φ(x) for x^{0} < y^{0}
Therefore,
G(x,y) = <0|Θ(x^{0} - y^{0})φ(x)φ^{†}(y) + Θ(y^{0} - x^{0})φ^{†}(y)φ(x)|0>
The propagator of the complex scalar field can be calculated in
the same fashion as for the non-complex case. This can be
seen from:
<φ(x)φ^{†}(y)> = (1/2)[<φ_{R}(x)φ_{R}(y) - iφ_{R}(x)φ_{L}(y) + φ_{L}(x)φ_{R}(y) + φ_{L}(x)φ_{L}(y)>]
^{ } = (1/2)[<φ_{R}(x)φ_{R}(y) + φ_{L}(x)φ_{L}(y)>]
^{ } = <φ(x)φ(y)>
The resulting Feynman propagator in 4-dimensional Minkowski
spacetime is:
G(x,y) = -∫(d^{4}p/(2π)^{4}) exp(-ip(x-y))/(p^{2} - m^{2} + iε)
The interpretation now is that the amplitude for the particle to
propagate from y -> x cancels the amplitude for the antiparticle
to travel from x -> y. In fact, this interpretation is also there
for the real scalar field because, as stated, the particle is its own
antiparticle.
As before, the propagator is equal to the Green's function of the
non-complex case.
As stated previously, the Fourier transform of the position space
propagators can be thought of as propagators in momentum space.
G_{F}(p) = -i/(p^{2} - m^{2} + iε)
Massive Fermionic Fields
------------------------
The Dirac equation is:
(iγ^{μ}∂_{μ} - m)ψ = 0
The simplest solutions to the DE are plane waves u(p)exp(-ikx) and
v(p)exp(ikx) where u(p) and v(p) are Dirac spinors.
The Dirac Lagrangian is (h = c = 1):
_
L = ψ(iγ^{μ}∂_{μ} - m)ψ
_
L = ψ(iγ^{0}∂ψ/∂t + iγ^{0}γ^{j}/γ^{0}∂ψ/∂x - mψ)
_
ψ = γ^{0}ψ^{†}. The ψ^{†} by itself does not result in a Lorentz invariant
Lagrangian and Lorentz invariant Dirac equation after the Euler-
Lagrange equations have been applied to minimize the action.
Although, the Dirac equation is first order in derivatives, it is
the magic of the γ matrices that makes the Lagrangian Lorentz
invariant!
_
ψ(x) creates a Fermion. ψ(x) annihilates a Fermion.
The canonical momenta are:
. _
π = ∂L/∂ψ = iψγ^{0} = iγ^{0}ψ^{†}γ^{0} = iψ^{†} (γ^{0})^{2} = 1)
. _
π^{†} = ∂L/∂ψ^{†} = iψγ^{0} = iγ^{0}ψ^{†}γ^{0} = 0
Therefore:
{ψ(x),π(y)} ≡ {ψ(x),iψ^{†}(y)}
Now,
{ψ(x),iψ^{†}(y)} = ψ(x)iψ^{†}(y) + iψ^{†}(y)ψ(x)
_{ } = i(ψ(x)ψ^{†}(y) + ψ^{†}(y)ψ(x))
_{ } = i{ψ(x),ψ^{†}(y)}
The anticommutator obeys the same rule as the commutator in that:
{ψ(x),π(y)} = ihδ(x - y)
Therefore, by comparison:
{ψ(x),ψ^{†}(y)} = hδ(x - y)
The field operator equations are:
ψ(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω)Σ_{s}[b_{p}^{s}u^{s}(p)exp(ip.x) + c_{p}^{s†}v^{s}(p)exp(-ip.x)]
ψ^{†}(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω)Σ_{s}[b_{p}^{s†}u^{s†}(p)exp(-ip.x) + c_{p}^{s}v^{s†}(p)exp(ip.x)]
π(x) = iψ^{†} = ∫(d^{3}p/(2π)^{3}) (i/√(2ω)Σ_{s}[b_{p}^{s†}u^{s†}(p)exp(-ip.x) + c_{p}^{s}v^{s†}(p)exp(ip.x)]
π^{†}(x) = 0
b_{p}^{s} creates a fermion of momentum p and spin s.
c_{p}^{s} creates an antifermion of momentum p and spin s.
We can summarize the anti-commutators for the fermionic field
as follows:
{b_{p}^{r},b_{q}^{s†}} = {c_{p}^{r},c_{q}^{s†}} = (2π)^{3}δ^{rs}δ^{(3)}(p - q)
{ψ(x)_{α},ψ(y)^{†}_{β}} = hδ_{αβ}δ(x - y)
All other combinations of b_{p}, b_{p}^{†}, c_{p}, c_{p}^{†} = 0
All other combinations of ψ(x) and ψ^{†}(x) = 0
Where r and s represents the spin and α and β are the spinor
indeces (u_{1}, u_{2}, v_{1}, v_{2}).
Conserved Quantities
--------------------
It is easy to see that the Lagrangian is invariant under the
_ _
transformation (φ -> exp(iθ)φ and φ -> φexp(-iθ)). This implies
there is a symmetry and a conserved quantity. We can derived
the conserved quantity in 2 ways. Start with the Dirac equation:
(iγ^{μ}∂_{μ} + m)ψ = 0 ... 1.
Which expands to be:
(iγ^{0}∂_{0}ψ + iγ^{i}∂_{i}ψ + mψ) = 0
The Hermitian conjugate is:
(i∂_{0}ψ^{†}γ^{0†} + i∂_{i}ψ^{†}γ^{i†} - mψ†) = 0
Now, γ^{0†} = γ^{0} and γ^{i†} = -γ^{i}
Therefore,
(i∂_{0}ψ^{†}γ^{0} + i∂_{i}ψ^{†}(-γ^{i}) - mψ†) = 0
Multiply by γ^{0} from the right and make use of the defining
relationship:
γ^{0}γ^{i} + γ^{i}γ^{0} = 2η_{0i}I = 0 ∴ γ^{0}γ^{i} = -γ^{i}γ^{0}
To get:
^{ } _ ^{ } _^{ } _
(i∂_{0}ψγ^{0} + ∂_{i}ψiγ^{i} - mψ) = 0
Or,
_^{ }
ψ(i∂_{μ}γ^{μ} - m) = 0 ... 2.
_
Where ψ = γ^{0}ψ^{†}
This is the ADJOINT DIRAC EQUATION.
_
Multiplying equation 1. by ψ from the left and equation 2 by
ψ from the right and adding gives:
_^{ } _
ψ(iγ^{μ}∂_{μ} + m)ψ + ψ(i∂_{μ}γ^{μ} - m)ψ = 0
_^{ } _
ψ(γ^{μ}∂_{μ}ψ) + (ψ∂_{μ}γ^{μ})ψ = 0
Now, the 2nd term on the LHS is interpreted as being right to
left so this is equivalent to:
_ ^{ } _^{ } _
ψγ^{μ}∂_{μ}ψ + (∂_{μ}ψ)γ^{μ}ψ = ∂_{μ}(ψγ^{μ}ψ) = ∂_{μ}(j_{V}^{μ}) = 0
Again, this is the CONTINUITY EQUATION where j_{V}^{μ} is the 4-VECTOR
PROBABILITY CURRENT.
The PROBABILITY DENSITY is defined as:
^{ } _
ρ = j_{V}^{0} = ψγ^{0}ψ
Therefore we can write,
j_{V}^{μ} = (ρ,j_{V}^{i})
The other way to get at the conserved quantity is similar to the
method already encountered with the complex scalar field. We
can write:
_
L = ψ(iγ^{μ}∂_{μ} + m)ψ
_^{ } _
= iψγ^{μ}∂_{μ}ψ + mψψ
The canonical momentum is:
. _
π = ∂L/∂ψ = iψγ^{0}
_ _
π = ∂L/∂(∂_{0}ψ) = 0
The transformations are:
ψ -> ψ - iεψ -> ψ - εf where f = -iψ
_ _ _ _ _ _
ψ -> ψ + iεψ -> ψ + εf where f = iψ
Look for a conserved quantity using Noether's Theorem.
__
Conserved quantity = ∫[πf + πf]d^{3}x
_
= ∫[iψγ^{0}(-iψ)]d^{3}x
_
= ∫[ψγ^{0}ψ]d^{3}x
Again, the quantity in [] is the charge density of the field, ρ.
∴ Q = ∫ρd^{3}x
Substituting the equations for the field operators yields:
ψ(x) = ∫ d^{3}p Σ_{s}[c_{p}^{†}^{s}c_{p}^{s} - d_{p}^{s†}d_{p}]
Axial Vector Current
--------------------
When m = 0, the Dirac Lagrangian also has an extra internal symmetry
which rotates left and right-handed fermions in opposite directions.
We can go through a similar procedure to the above to get:
ψ -> exp(iαγ^{5})ψ
_ _
ψ -> ψexp(iαγ^{5})
^{ } _
j_{A}^{μ} = ψγ^{5}γ^{μ}ψ
The Fermionic Propagator
------------------------
For a Dirac field:
_
S(x,y) = <0|Tψ(x)ψ(y)|0>
Where,
_
Tψ(x)ψ(y) = ψ(x)ψ(y) for x^{0} > y^{0}
_
= -ψ(y)ψ(x) for x^{0} < y^{0}
The - sign is required for Lorentz invariance.
Therefore,
_{ } _ _
S(x,y) = <0|Θ(x^{0} - y^{0})ψ(x)ψ(y) + Θ(y^{0} - x^{0})ψ(y)ψ(x)|0>
The propagator of the Dirac field can be calculated in the same
fashion as before.
S_{F}(x,y) = -∫(d^{4}p/(2π)^{4}) γ^{μ}p_{μ}exp(-ip(x-y))/(p^{2} - m^{2} + iε)
Where γ^{μ}p_{μ} = γ^{0}p_{0} + γ^{j}p_{j}
This satisfies:
(iγ^{μ}∂_{μ} + m)S_{F}(x,y) = iδ^{(4)}(x - y)
So S_{F}(x,y) is a Green's function for the Dirac operator.
As stated previously, the Fourier transform of the position space
propagator can be thought of as propagators in momentum space.
S_{F}(p) = (γ^{μ}p_{μ} + m)/(p^{2} - m^{2} + iε)
Massless Spin 1 Vector Fields (The Electromagnetic Field)
---------------------------------------------------------
There is no equivalent equation to the Klein-Gordon, Dirac
or Schrodinger equations that describes the wavefunction
of the photon. This is because photons always behave
relativistically and can be freely emitted and absorbed.
Instead, a single photon is described quantum mechanically
by Maxwell's equations, where the solutions are taken to be
complex.
The electromagnetic 4 potential is defined as:
A_{μ} = (φ/c, -A)
Where φ is the electric scalar potential and A is the magnetic
vector potential. The B and E fields in terms of φ and A are:
B = ∇ x A
E = -∇φ - ∂A/∂t
We can also define an asymmetric tensor, F_{μν} such that,
F_{μν} = ∂_{μ}A_{ν} - ∂_{ν}A_{μ}
- -
| 0 E_{x}/c E_{y}/c E_{z}/c |
|-E_{x}/c 0 -B_{z} B_{y} | = F_{μν}
|-E_{y}/c B_{z} 0 -B_{x} |
|-E_{z}/c -B_{y} B_{x} 0 |
- -
- -
| 0 -E_{x}/c -E_{y}/c -E_{z}/c |
| E_{x}/c 0 -B_{z} B_{y} | = F^{μν}
| E_{y}/c B_{z} 0 -B_{x} |
| E_{z}/c -B_{y} B_{x} 0 |
- -
From these we get Maxwell's Equations in the absence of charge
sources:
∇.B = 0
∂B/∂t = -∇ x E
∇.E = 0
∂E/∂t = -∇ x B
The Lagrangian for Maxwell's equations is:
L = (-1/4)F_{μν}F^{μν} - J^{μ}A_{μ}
In the absence of charge sources is this becomes:
L = (-1/4)F_{μν}F^{μν} ≡ (-1/2)(B^{2} - E^{2})
= (-1/2)(∂_{μ}A_{ν}∂^{μ}A^{ν} - ∂_{ν}A_{μ}∂^{μ}A^{ν})
Substituting this into the Euler-Lagrange equations for a field gives:
∂_{μ}(∂L/∂(∂_{μ}A_{ν}) = ∂L/∂A_{ν}
_{ } = 0
So the E-L equation becomes:
∂_{μ}(∂^{μ}A^{ν} - ∂^{ν}A^{μ}) = 0
∂_{μ}∂^{μ}A^{ν} - ∂_{μ}∂^{ν}A^{μ} = 0
∂_{μ}F^{μν} = 0
Now, the electromagnetic field is a vector field that can be decomposed
into longitudinal and transverse components. Longitudinal components
do not have a curl and are parallel to the direction of motion. Transverse
components do not have divergence and are perpendicular to the direction
of motion. Thus,
∇.F_{T} = 0
and
∇ x F_{L} = 0
This is referred to as the HELMHOLTZ DECOMPOSITION. Since ∇.E = ∇.B = 0
in the absence of sources, both E and B are transverse.
Gauge Fixing
------------
Different representative configurations of a physical state are called
different gauges. Picking a gauge is rather like picking coordinates
that are adapted to a particular problem. Moreover, different gauges
often reveal slightly different aspects of a problem. Since the
physical fields E and B are not modified by a gauge transformation,
we may say that the potentials φ and A contain both a physical part
(because the physical fields can be computed from them) and a nonphysical
part (which changes when we do a gauge transformation). We will look at
2 gauges: the Coulomb gauge and the Lorentz gauge. Choosing the Coulomb
gauge makes A into a transverse field that correctly restricts the photon
to 2 degrees of freedom corresponding to the 2 polarization states.
However the field commutation relations are a little more involved as is
the propagator. The Coulomb gauge is also not Lorentz invariant although
the final result is. Choosing the Lorentz gauge results in 4 degrees of
freedom that need to be dealt with, but the field commutators follow the
familiar form and the propagator is simpler. The Lorentz gauge is also
manifestly Lorentz invariant from the outset.
Coulomb Gauge
-------------
E = -∇φ - ∂A/∂t. Therefore,
∇.E = -∇^{2}φ - (∂(∇.A)/∂t) = 0
In the Coulomb gauge ∇.A = 0 so the equation of motion becomes:
∇^{2}φ = 0
Let us pause for a moment to look at this. A is a 4-vector with the
time component set to 0 (in other words a 3-vector). This allows
the remaining part of the potential to satisfy the condition ∇.A = 0
(≡ ∂^{i}A_{i} = 0).
We can also use ∂_{μ}∂^{μ}A^{ν} - ∂_{μ}∂^{ν}A^{μ} = 0 to derive the equations of motion.
The second term is ∂^{ν}(∇.A) = 0 leaving the first term ∂_{μ}∂^{μ}A^{ν} ≡ ∇^{2}A = 0.
The momentum conjugate to A_{μ} is:
.
π^{0} = ∂L/∂A_{0} = 0
.
π^{i} = ∂L/∂A_{i} = -F^{0i} = E^{i}
The field operator equations are similar to the real scalar field:
A(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x) + a_{p}^{r†}exp(-ip.x)]
and
π^{i} = E(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))(-iω)Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x) - a_{p}^{r†}exp(-ip.x)]
^{ } = ∫(d^{3}p/(2π)^{3}) (-i√(ω/2))Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x) - a_{p}^{r†}exp(-ip.x)]
Where ξ^{r} are the 2 POLARIZATION VECTORS. These polarization
vectors obey the rules ξ_{1}.ξ_{2} = ξ_{1}.p = ξ_{2}.p = 0 and ξ_{1}^{2} = ξ_{2}^{2} = 1 and
thereby satisfy the transversality condition imposed by the gauge.
r = 1 to 2.
We can summarize the commutators for the electromagnetic field
as follows:
[a_{p}^{r},a_{q}^{s†}] = (2π)^{3}δ^{rs}δ^{(3)}(p - q)
[a_{p}^{r},a_{q}^{s}] = [a_{p}^{r†},a_{q}^{s†}] = 0
Which is the same as the usual commutation rules for creation
and anihilation operators. However, the field commutation rules
are a little more complicated.
[A_{i}(x),E_{j}(y)] = ihδ_{⊥}^{ij}(x - y)
_{ } = ih(δ_{ij} - p_{i}p_{j}/p^{2})δ^{(3)}(x - y)
_{ } = ih(δ_{ij} - ∂_{i}∂_{j}/∇^{2})δ^{(3)}(x - y) since p = -i∂/∂x
[A_{i}(x),A_{j}(y)] = [E_{i}(x),E_{j}(y)] = 0
Where δ^{⊥}_{ij}(x - y) is the TRANSVERSE DELTA FUNCTION. The TDF
acts like a δ function on fields with 0 divergence and projects
an arbitrary vector field onto its transverse part. The
projection is most easily constructed in Fourier space.
F_{T}(x) = ∫(d^{3}p/(2π)^{3}) (δ_{ij} - p_{i}p_{j}/p^{2}) F_{L}(p) exp(ipx)
The term in () also satisfies the following completeness
relationship.
Σξ_{i}^{r}ξ_{j}^{r} = (δ_{ij} - p_{i}p_{j}/p^{2})
Projection Operators and Completeness
-------------------------------------
Recall from the discussion of projection operators in Quantum
Mechanics (Bell's Theorem).
I = Σ_{n}|i><i| = Σ_{n}P_{i} where P_{i} is the projection onto i.
Also recall from QM the completeness relationship.
ψ(x) = Σ_{n}a_{n}ψ_{n}(x) where a_{n} = ∫dx' ψ_{n}^{*}(x')ψ(x')
Therefore,
ψ(x) = Σ_{n}(∫dx' ψ_{n}^{*}(x')ψ(x'))ψ_{n}(x)
= ∫dx' (Σ_{n}(ψ_{n}^{*}(x')ψ_{n}(x))ψ(x')
This is an equation of the form:
ψ(x) = ∫dx' G(x,x')ψ(x') where G is Green's function.
Consider the eigenvalue equation
D|ψ_{n}> = λ_{n}|ψ_{n}> where D is a linear differential operator.
We can write:
DG(x,x') = λ_{n}G(x,x')
Where G(x,x') is a Green's Function such that:
DG(x,x') ≡ λ_{n}G(x,x') = δ(x - x')
Now λ_{n} is just a scalar so we can write
λ_{n}ψ(x) = λ_{n}∫dx' δ(x - x') ψ(x')
By comparison we see that:
δ(x - x') = Σ_{n}(ψ_{n}^{*}(x')ψ_{n}(x))
Returning to the completeness relationship for the polarization
vectors we see:
Σξ_{i}^{r}ξ_{j}^{r} ≡ Σ|ξ_{i}^{r}><ξ_{j}^{r}|
= |ξ_{i}^{1}><ξ_{j}^{1}| + |ξ_{i}^{2}><ξ_{j}^{2}|
This is a projection onto the sub space spanned by the polarisation
vectors. This sub space is orthogonal to the momentum.
The Lorentz Gauge
-----------------
We now take time into consideration and replace 3-vectors with
4-vectors associated with Minkowski spacetime.The Lorentz
gauge is manifestly Lorentz invariant and is defined as:
∇.A + ∂φ/∂t = 0
Which can be written in space-time as:
∂^{μ}A_{μ} = 0
The equation of motion is found from:
∂_{μ}∂^{μ}A^{ν} - ∂_{μ}∂^{ν}A^{μ} = 0
Which, after imposing the gauge becomes:
∂_{μ}∂^{μ}A^{ν} = 0
These equations of motion can also be derived from the following
Lagrangian:
L = (-1/4)F_{μν}F^{μν} - (1/2)(∂_{μ}A^{μ})^{2}
The momentum conjugate to A_{μ} is:
.
π^{0} = ∂L/∂A_{0} = -∂_{μ}A^{μ}
.
π^{i} = ∂L/∂A_{i} = -F^{0i} = E^{i}
The field operator equations are once again:
A_{μ}(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x) + a_{p}^{r†}exp(-ip.x)]
and
E_{μ}(x) = ∫(d^{3}p/(2π)^{3}) (-i√(ω/2))Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x) - a_{p}^{r†}exp(-ip.x)]
But this time since there is no restriction regarding transverse
and longitudinal components, ξ^{r} now represents 4 polarization
4-vectors versus 2 3-vector polarization vectors in the Coulomb
gauge. These can be characterized as 2 tranverse, 1 longitudinal
and 1 time-like. Therefore, r = 0 to 3.
We can summarize the commutators in this gauge as follows:
[a_{p}^{μ},a_{q}^{ν†}] = -(2π)^{3}g^{μν}δ^{(3)}(p - q)
[a_{p}^{μ},a_{q}^{ν}] = [a_{p}^{μ†},a_{q}^{ν†}] = 0
g_{μν} is necessary to achieve Lorentz invariance.
The -ve sign is problematic because it implies negative probabilities
in the Hilbert space we are working in. The minus is entirely due to
the time component (note π^{0} = -∂_{μ}A^{μ}. The challenge is to 'fix' this
problem and also deal with the 2 'extra' degrees of freedom that we
have. The solution is to impose the gauge condition ∂^{μ}A_{μ} = 0. However,
if we impose the gauge condition at the Lagrangian level we end up
where we started in the Coulomb gauge. The answer is to impose this
as a constraint on the physical states associated with the tranvese
photon. This is done by splitting A_{μ}(x) into 2 separate operators.
A_{μ}(x) = A_{μ}^{+}(x) + A_{μ}^{-}(x)
Therefore,
∂^{μ}A_{μ}(x) = ∂^{μ}A_{μ}^{+}(x) + ∂^{μ}A_{μ}^{-}(x)
Where,
∂^{μ}A_{μ}^{+}(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))(ip)Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x)] represents the +ve
frequency (annhilation).
and
∂^{μ}A_{μ}^{-}(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))(-ip)Σ_{r}ξ^{r}[a_{p}^{r†}exp(-ip.x)] represents the -ve
frequency (creation)
We then impose the Lorentz condition on the physical states through
the eigenvalue equations:
∂^{μ}A_{μ}^{+}|Ψ> = 0|Ψ and <Ψ|∂^{μ}A_{μ}^{-} = <Ψ|0>
This ensures the condition <Ψ|∂^{μ}A_{μ}|Ψ> = 0 so that the expectation
value of ∂^{μ}A_{μ} is 0.
This is referred to as the GUPTA-BLEULER condition.
Now, we want get rid of the 0 and 3 components since we know
these are 'unphysical' states. Let us write:
∂^{μ}A_{μ}^{+}|Ψ> = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))(ip)Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x)]|Ψ> = 0
We can write p.ξ^{1} = p.ξ^{2} = 0 for the transverse components and
p.ξ^{0} = -p.ξ^{3} for the time and longitudinal components.
Thus, we end up with
∫(d^{3}p/(2π)^{3}) (1/√(2ω))(ip)Σ_{r}ξ^{r}[(a_{p}^{3} - a_{p}^{0})exp(ip.x)]|Ψ> = 0
Where r is now 0 and 3. We immediately conclude that:
(a_{p}^{3} - a_{p}^{0})|Ψ> = 0
which leads to the crucial identity:
<Ψ|a_{p}^{3†}a_{p}^{3}|Ψ> = <Ψ|a_{p}^{0†}a_{p}^{0}|Ψ>
So only the transverse modes survive and contribute to the dynamics.
Therefore, it is consistent to implement the Lorenz gauge condition and
indeed decouples the unwanted degrees of freedom from the physical
Hilbert space.
With this condition applied the commutation relations are restored to
their familiar form:
[a_{p}^{μ},a_{q}^{ν†}] = (2π)^{3}g^{μν}δ^{(3)}(p - q)
[a_{p}^{μ},a_{q}^{ν}] = [a_{p}^{μ†},a_{q}^{ν†}] = 0
[A_{μ}(x),E_{ν}(y)] = ihg_{μν}δ^{(3)}(x - y)
[A_{μ}(x),A_{ν}(y)] = [E_{μ}(x),E_{ν}(y)] = 0
The Photon Propagator
---------------------
In the Coulomb gauge the propagator is:
G(x,y) = <0|TA_{μ}(x)A_{ν}(y)|0>
= ∫(d^{4}p/(2π)^{4}) i/(p^{2} + iε)(δ_{ij} - p_{i}p_{j}/p^{2})exp(-ip.(x-y))
In the Lorentz gauge the propagator is:
G(x,y) = <0|TA_{μ}(x)A_{ν}(y)|0>
= ∫(d^{4}p/(2π)^{4}) i/(p^{2} + iε)exp(-ip.(x-y))
As stated previously, the Fourier transform of the position space
propagators can be thought of as propagators in momentum space.
Thus, in the Lorentz gauge we get:
G(p) = -ig_{μν}/(p^{2} + iε)
Massive Spin 1 Vector Fields
----------------------------
In the discussion of Quantum Field theory we have talked about real
and complex scalar fields, fermionic fields and the electromagnetic
field. The scalar fields represent massive spin 0 particles, the
fermionic field represents massive half-integer spin particles and
the electromagnetic field represents massless spin 1 particles.
The question is "how do we represent massive spin 1 particles?".
The W^{±} and Z^{0} gauge bosons are examples of particles (also called
"vector bosons") that fall into this category.
It seems logical that somehow we should try and incorporate mass
into the spin 1 electromagnetic vector field. However, simply adding
a mass term to Lagrangian as follows spoils the invariance.
L = (-1/4)F_{μν}F^{μν} - (1/2)m^{2}A_{μ}A^{μ}
This is discussed here
However, there is a way to include an extra field such that the
resulting Lagrangian invariant and includes a massive vector field.
Consider the Lagrangian
L = (-1/4)F_{μν}F^{μν} - (1/2)(∂_{μ}φ + mA_{μ})(∂^{μ}φ + mA^{μ})
Where φ is a real scalar field.
This is the STUECKELBERG Lagrangian. It is an extension to the
Standard Model that provides a mechanism for mass generation
in an Abelian gauge theory while preserving gauge invariance. In
essence it provides a mass to the physical photon. It is a special
case of the Higgs mechanism where the Higgs excitations are so
large they can be ignored.
The procedure for quantizing the field is the same as we have
previously encountered. The math is similar and will not be
repeated here.
As a fotnote it is worth noting that while the Stueckelberg 'trick'
works for the Abelian case, the Higgs mechanism involving
spontaneous symmetry breaking remains the only presently known
way to give masses to non-Abelian vector fields.
Massive Vector Field Propagator
--------------------------------
Like the electromagnetic field, the propagator for a massive
vector field is dependant on the gauge chosen. In the Feynman
gauge this is
G(p) = g_{μν}/(p^{2} - m^{2} + iε)
The Field Hamiltonians
----------------------
The Legendre transformation yields:
.
H = πφ - L
.
π = ∂L/∂φ
H = ∫d^{3}x H'
Real Scalar
-----------
L = (1/2)∂_{μ}φ∂^{μ}φ - (1/2)m^{2}φ^{2}
= (1/2)∂^{2}φ/∂t^{2} + (1/2)∇^{2}φ - (1/2)m^{2}φ^{2}
. .
π = ∂L/∂φ = φ
.
H' = πφ - L
. .
= φ^{2} - (1/2)φ^{2} - (1/2)(∂φ/∂x)^{2} - (1/2)m^{2}φ^{2}
.
= (1/2)φ^{2} - (1/2)(∂φ/∂x)^{2} - (1/2)m^{2}φ^{2}
Substituting the expression for φ and π and noting that ω^{2} = p^{2} + m^{2}
φ(x) = ∫(d^{3}p/(2π)^{3}√(2ω)) (1/√(2ω))[a_{p}^{†}exp(-ipx) + a_{p}exp(ipx)]
and
π(x) = (-i/2)∫(d^{3}p/(2π)^{3} [a_{p}^{†}exp(-ipx) - a_{p}exp(ipx)]
Leads to:
H_{renorm}' = ∫d^{3}p/(2π)^{3} ω_{p} a_{p}^{†}a_{p}
The order of the creation operators is important to achieve a vacuum
expectation value of 0. The creation and annihilation must satisfy
<0|a_{p}^{†} = 0 and|a_{p}|0> = 0. This requires that when taking the product
of quantum fields, or equivalently their creation and annihilation
operators, all creation operators are placed to the left of all
annihilation operators in the product. This is referred to as NORMAL
ORDERING. The normal ordered string of operators is written as:
:a_{p}^{†}a_{p}a_{p}^{†}a_{p}: = a_{p}^{†}a_{p}^{†}a_{p}a_{p}
Note also that the ∫ -> ∞ unless a cutoff in the momentum is imposed.
This is discussed in the section on Feynman diagrams.
Complex Scalar Field
--------------------
Using a similar procedure to the above we get:
H_{renorm}' = (1/2)∫d^{3}p/(2π)^{3} ω_{p} (b_{p}^{†}b_{p} + c_{p}^{†}c_{p})
Schrodinger
-----------
L = ψ^{†}(i∂/∂t + ∇^{2}/2m)ψ
^{ } .
= iψ^{†}ψ + ψ^{†}∇^{2}ψ/2m
.
π = ∂L/∂ψ = iψ^{†}
.
H' = πψ - L
^{ } .^{ } .
= iψ^{†}ψ - iψ^{†}ψ + ψ^{†}∇^{2}ψ/2m
= ψ^{†}(-∇^{2}/2m)ψ
This is just the expectation value of the operator.
<H> = <φ|H|φ>
= ∫d^{3}x ψ^{†}[p^{2}/2m]ψ
Now,
H|ψ_{m}> = E_{m}|ψ_{m}>
_{ } = ω_{m}|ψ_{m}> (h = 1)
E = ∫dx ψ^{†} ω ψ
= ∫d^{3}x ω Σ_{m}a_{m}^{†}ψ_{m}^{†} Σ_{n}a_{n}ψ_{n}
= ∫d^{3}x ω Σ_{mn}a_{m}^{†}a_{n}ψ_{m}^{†}ψ_{n}
= ∫d^{3}x ω Σ_{mn}a_{m}^{†}a_{n}δ_{mn}
= ∫d^{3}x ω Σ_{p}a_{p}^{†}a_{p}
= ∫d^{3}x d^{3}/(2π)^{3} ω_{p} a_{p}^{†}a_{p}
= ∫d^{3}x H'
Fermionic
---------
_
L = Ψ(iγ^{μ}∂_{μ} - m)Ψ
_
= Ψ(iγ^{0}∂Ψ/∂t + iγ^{0}γ^{i}/γ^{0}∂Ψ/∂x - mΨ)
_ . ^{ } _ _
= Ψiγ^{0}Ψ + Ψiγ^{i}∇Ψ - ΨmΨ
. _
π = ∂L/∂Ψ = iγ^{0}Ψ
.
H' = πΨ - L
^{ } _. _ ^{ }. _ ^{ } _
= iγ^{0}ΨΨ - Ψiγ^{0}Ψ - Ψiγ^{i}∂_{i}Ψ + ΨmΨ
_ ^{ } _
= -Ψiγ^{i}∂_{i}Ψ + ΨmΨ
_
= Ψ(-iγ^{i}∂_{i} + m)Ψ
This is just the expectation value of the operator (-iγ^{i}∂_{i} + m)
Consider:
(iγ^{i}∂_{i} - m)Ψ = 0
Rewrite as:
(iγ^{0}∂_{0} + iγ^{i}∂_{i} - m)u(p)exp(-ipx) = 0
Or, since p_{0} = i∂/∂t and p_{i} = i∂/∂x,
(γ^{0}p_{0} + γ^{i}p_{i} - m)u(p)exp(-ipx) = 0
Rearranging,
γ^{0}p_{0}u(p)exp(ip.x) = (-γ^{i}p_{i} + m)u(p)exp(ip.x)
and
-γ^{0}p_{0}v(p)exp(-ip.x) = (γ^{i}p_{i} + m)v(p)exp(-ip.x)
Now,
Ψ = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))Σ_{s}[b_{p}^{s}u^{s}(p)exp(ip.x) + c_{p}^{s†}v^{s}(p)exp(-ip.x)]
Which leads to:
(-iγ^{i}∂_{i} + m)Ψ = ∫(d^{3}p/(2π)^{3}) (1/√(2ω)) Σ_{s}[b_{p}^{s}(-iγ^{i}∂_{i} + m)u^{s}(p)exp(ip.x)
+ c_{p}^{s†}(iγ^{i}∂_{i} + m)v^{s}(p)exp(-ip.x)]
Which leads to:
(-iγ^{i}∂_{i} + m)Ψ = ∫(d^{3}p/(2π)^{3}) (1/√(2ω)) Σ_{s}[b_{p}^{s}(γ^{0}p_{0})u^{s}(p)exp(ip.x)
- c_{p}^{s†}(γ^{0}p_{0})v^{s}(p)exp(-ip.x)]
(-iγ^{i}∂_{i} + m)Ψ = ∫(d^{3}p/(2π)^{3}) (1/√(2ω)) (γ^{0}p^{0}) Σ_{s}[b_{p}^{s}u^{s}(p)exp(ip.x)
- c_{p}^{s†}v^{s}(p)exp(-ip.x)]
Now from Special Relativity p_{0} is equivalent to the energy E (= ω).
Thus,
(-iγ^{i}∂_{i} + m)Ψ = ∫(d^{3}p/(2π)^{3}) (√(ω/2)) γ^{0} Σ_{s}[b_{p}^{s}u^{s}(p)exp(ip.x)
- c_{p}^{s†}v^{s}(p)exp(-ip.x)]
Now,
Ψ^{†}(x) = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))Σ_{s}[b_{p}^{s†}u^{s†}(p)exp(-ip.x) + c_{p}^{s}v^{s†}(p)exp(ip.x)]
Which leads to:
Ψ^{†}(-iγ^{i}∂_{i} + m)Ψ = ∫(d^{3}pd^{3}q/(2π)^{6}) (√(ω/2))(1/√(2ω)) γ^{0} (Σ_{r}[b_{p}^{r}u^{r}(p)exp(ip.x)
- c_{p}^{r†}v^{r}(p)exp(-ip.x)]
Σ_{s}[c_{q}^{s†}u^{s†}(q)exp(-iq.x) + d_{q}^{s}v^{s†}(q)exp(iq.x)])
Ψ^{†}(-iγ^{i}∂_{i} + m)Ψ = ∫(d^{3}pd^{3}q/(2π)^{6}) (1/2) γ^{0} (Σ_{r}[b_{p}^{r}u^{r}(p)exp(ip.x)
- c_{p}^{r†}v^{r}(p)exp(-ip.x)]
Σ_{s}[c_{q}^{s†}u^{s†}(q)exp(-iq.x) + d_{q}^{s}v^{s†}(q)exp(iq.x)])
_
Now Ψ = γ^{0}Ψ^{†} and (γ^{0})^{2} = 1
Therefore,
_
Ψ(-iγ^{i}∂_{i} + m)Ψ = ∫(d^{3}pd^{3}q/(2π)^{6}) (1/2) (Σ_{r}[b_{p}^{r}u^{r}(p)exp(ip.x)
- c_{p}^{r†}v^{r}(p)exp(-ip.x)]
Σ_{s}[c_{q}^{s†}u^{s†}(q)exp(-iq.x) + d_{q}^{s}v^{s†}(q)exp(iq.x)])
After manipulation and making use of the inner product relationship
between u and v we end up with:
H' = ∫(d^{3}/(2π)^{3}) ω(b_{p}^{s†}b_{p}^{s} + c_{p}^{s†}c_{p}^{s})
Electromagnetic
---------------
From the electromagnetic tensor we get:
L = (-1/4)F_{μν}F^{μν} = (-1/2)(B^{2} - E^{2})
In the Coulomb gauge the canonical momenta are:
π^{0} = ∂L/∂A_{0} = 0
.^{ } . . .
π^{i} = ∂L/∂A_{i} = A_{i} = E^{i} (same as real field with φ -> A)
H' = πA_{i} - L
.
H' = (A_{i})^{2} - L
= E^{2} + (1/2)(B^{2} - E^{2})
= (1/2)E^{2} + (1/2)B^{2}
We can also get this another way:
.
H' = (1/2)A^{2} + (1/2)(∂A/∂x)^{2} + (1/2)m^{2}A^{2}
Now m = 0 so ω^{2} = p^{2}
.
H' = (1/2)A^{2} + (1/2)(∂A/∂x)^{2}
Now,
A = ∫(d^{3}p/(2π)^{3}) (1/√(2ω))Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x) + a_{p}^{r†}exp(-ip.x)]
Therefore,
∂A/∂x =∫(d^{3}p/(2π)^{3}) (i/√(2ω))(ip) Σ_{r}ξ^{r}[a_{p}^{r}exp(ip.x) - a_{p}^{r†}exp(-ip.x)]
Now p = -i∇ ∴ ip = ∇ and B = ∇ x A. Therefore, B = ip x A and we
can also write:
B(x) = ∫(d^{3}p/(2π)^{3}) (i/√(2ω))Σ_{r}{∇ x ξ^{r}[a_{p}^{r}exp(ip.x) - a_{p}^{r†}exp(-ip.x)]}
Therefore,
(∂A/∂x)^{2} ≡ B^{2}
So we again get:
H' = (1/2)E^{2} + (1/2)B^{2}
After substituting in the expressions for E and B we get:
H' = ∫(d^{3}p/(2π)^{3}) ω Σ_{r}a_{p}^{r†}a_{p}^{r}
Where r = 1 -> 2
Note: In the Lorentz gauge we would get:
H' = ∫(d^{3}p/(2π)^{3}) ω Σ_{r}a_{p}^{r†}a_{p}^{r}
Where r = 0 -> 3
However, the 0 and 3 components cancel leaving the Coulomb result.