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Quantum Field Theory Notes

Lorentz transformations The definition of a Lorentz transformation is that it preserves the metric:

The three Lorentz rotations mix two spatial dimensions, while the three Lorentz boosts mix time with a spatial

dimension. We have the explicit matrices for the rotations:

Another way of writing this is:

With infinitesimal rotations:

Spinors transform under a slightly different rule:

With infinitesimal rotations:

The following table summarises transformation properties under these transformations:

Object Transform

Scalar

Vector 2-Tensor

Spinor

Classical field theory Suppose we have the Lagrangian:

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The conjugate field is then:

The Hamiltonian is given by the Legendre transformation:

This essentially means we substitute out for some function of and , and so get:

Field theories have potential terms (one field), propagating kinetic terms (two fields or derivatives of fields), and

interaction terms (three or more fields).

Euler-Lagrange equations The action is defined as:

The Lagrangian formulation posits that a field adopts the configuration that minimises the action with respect to

small changes in the field values over time or space. Thus we have:

This total derivative term is the integral of a 4-divergence over a 4-volume. This should go to zero at infinity so long

as the fields we are considering are locally confined (i.e. they go to zero at infinity). Then we are left with:

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Since this should hold over any arbitrary volume the square bracket term must be zero:

These are the Euler-Lagrange equations of motion that define how the field behaves.

Consider for example:

And so the equations of motion become:

Noether’s theorem If we have a continuous global (same everywhere) symmetry parameterised by , we can write:

If the fields satisfy their equation of motion then the first term is zero, leaving:

We thus define a conserved Noether current as:

Time evolution of states The S matrix is the evolution operator in the Heisenberg picture:

In the Heisenberg picture the field evolves according to:

Where is the time evolution operator that solves the SE for the field:

Considering a potential of the form:

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In the interaction picture fields only evolve according to , so they are free fields evolving as:

Combining this with the field evolution in the Heisenberg picture we get:

The operator thus relates the Heisenberg picture field to the free fields at some time . Differentiating

we get:

Using the result from before we have:

This has the solution:

This gives rise to the Dyson series expansion:

Matrix elements evolve as per:

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Quantisation of scalar fields The free scalar field is written as the sum over all momenta of momentum-dependent particle creation and

annihilation operators.

Quantisation of spin 1 fields For the probability interpretation of mechanics to hold and also the Lorentz invariance of relativity to be maintained,

we require that fields transform as unitary, irreducible representations of the Poincare group. Wigner showed in the

1930s that all such representations are infinite-dimensional, meaning that our basis will be momentum-dependent.

Proca Lagrangian: Ignores Gauge redundancy; massless version has infinite longitudinal polarisation vector, so

ultimately does not work as a theory.

The trouble is that in the massless case we have one fewer degree of freedom, as the longitudinal polarisation is

unphysical and does not propagate in this case. This yields an additional degree of freedom in the theory, leading to

Gauge redundancy. Gauge redundancy exists under the transformations:

Scalar QED Lagrangian: Here we take the massless Proca Lagrangian and try to add an interaction term between

and . To get this coupling to maintain Gauge invariance we need to define the covariant derivative:

We thus have the scalar QED Lagrangian:

The EM potential is quantised as:

The Ward identity The forward polarisation found in massless spin 1 equations is not physical, however it will mix with the two

transverse polarisations under Lorentz transforms – this is a problem. Thus we get terms like:

In terms of the matrix element this is:

This final term must to go zero if the theory is to work. We are thus led to the Ward identity:

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To check the ward identity, simply replace the incoming photon with and then show that .

In the case of the matrix element representing a fermion current this becomes:

In position space:

Which is an expression of the conservation of fermion current.

Unitarity and Lorentz invariance We need to describe QED fields using some representation of the Poincare group (Lorentz transformations and

translations in spacetime). This representation should be unitary to preserve the probability interpretation of QM:

Wigner classified all unitary, irreducible representations of the Poincare group, and they are all infinite dimensional.

They are classified by mass and spin. For , there are states for each (or 2 if m=0). For a vector

field of spin 1, there are thus 3 linearly independent polarisations for the massive case and 2 for the massless case.

This leads to gauge invariance in electromagnetism, and also the Ward identity in the massless case.

Consider a representation in a fixed basis . Now consider two possible ways

of defining the norms:

The first is positive definite and so satisfies probability interpretation while the second does not, but the first is not

Lorentz invariant. So we cannot satisfy both with a fixed basis!

Quantisation of spin ½ fields Terms in our Lagrangian must be Lorentz invariant under the spin ½ representation of the Lorenz group. In terms of

Weyl spinors and Pauli matrices such terms can be written:

Noting that and . If we now define Dirac spinors as:

And also define the gamma matrix as:

We can write this Lagrangian in the even more compact form:

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Which leads to the Dirac equation:

This equation has plane-wave spinor solutions:

Where and are the spinor polarisations for particles and antiparticles respectively. Solving the Dirac

equation in momentum space with the Weyl basis we have:

In the rest frame and this reduces to:

Solutions are then constants which we write as:

Which has four solutions that can be written:

Now considering a boosted frame with along the z axis we have:

Which has solutions:

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Which has four solutions that can be written:

Spinors are invariant under the gauge transformation:

In this case the covariant derivative takes the form:

We then have the fields:

Where creates antiparticles and

creates particles. Thus creates an antiparticle and annihilates a particle

at position .

The Dirac equation

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CPT invariance Charge conjugation is an operation on spinors that takes particles to antiparticles and flips the spin, as in:

Majorana fermions are their own charge conjugate, which means they do not carry a conserved charge. Terms like

, , and are all C invariant if we have:

The parity and time-reversal operations are defined as:

These are symmetries for QED but not for the weak interaction. Scalar fields under the parity operation transform as:

Photon vector fields have parity -1. In the Weyl basis, spinors transform as:

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This flips and so keeps spin invariant. By contrast, chiral theories (such as the theory of weak interactions)

are not parity invariant. We also have the electromagnetic potential transforming as:

Time reversal is the transformation . This means we must have , changing a positive

definite to a negative definite quantity, which cannot be done by a linear transformation. We must instead

implement it as an anti-linear transformation, with:

For T invariance to hold we must have the transformations:

QED as a whole is CPT invariant, a combination of all three transformations. The combined effect is equivalent to

that of sending particles to antiparticles that move as if in reverse in a mirror. Under CPT transforms we have:

Since all possible QED Lagrangian terms are CPT invariant, we have the result called the CPT theorem, which states

that any local Lorentz-invariant quantum field theory must have CPT symmetry.

Chirality and helicity Chirality refers to the handedness of a spinor – whether it is right or left handed. Helicity is the projection of spin

onto the direction of motion, and is equal to:

In the massless limit, the chirality states and are eigenstates of helicity.

Scalar Feynman propagator We begin with the simple two-point correlation function:

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Now for the time-ordered version:

Take in 1st term, which does not affect integral, leaving:

Using the result:

We have:

Photon propagator The equation of motion in the presence of a current takes the form:

In momentum space this is:

Adding a term to enforce gauge invariance we get:

Solving for the inverse operator (the Green’s function) we have:

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If we incorporate the effect of time ordering this becomes (similar method to the scalar case):

We thus arrive at the photon propagator:

In the Feynman gauge and we have the simpler form:

Dirac propagator We have:

From the anti-commutation relations for fermions we have:

Substituting this in we get:

Likewise:

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Taking out the momentum factors we have:

Now applying the time-ordering we get:

Using our result from before:

We can write:

This is more typically written in the form:

The cross section The cross section has units of area, and is a measure of interaction strength. It is defined as the probability of

interacting per unit flux per unit time:

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We here consider the case of a process, in which case in the center of mass frame we have

, and

hence:

The normalised differential probability is given by:

Where is the region of final momenta space being considered:

The normalisation factors are not simply 1, but are given by:

Putting the pieces together we get:

For convenience we define:

Substituting this into our equation we get at last:

Finding matrix elements Where we can decompose S into free theory and interacting theory parts:

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Matrix elements are given in terms of:

We find matrix elements using the LSZ reduction formula. In the case this is written as:

Applying a Dyson series expansion, we can write this in terms of the free fields and free vacuum:

Because the Dyson series has an infinite number of terms, we usually truncate to including only up to the first few

powers of the coupling constant ( is just a number that depends on the interaction):

Wick’s theorem tells us that each numerator term in the Dyson series expansion above term can be written as the

sum of all possible normal-ordered (all creation operators are on the left of annihilation operators) two-field

contractions, so for example:

Consider what happens in the case of three fields:

It turns out that when using Wick’s theorem to write the Dyson series in terms of Feynman propagators , all

propagators involving only integrated-over internal variables (e.g. or ) appear in both the numerator and

denominator, and so cancel out. We can thus simplify the Dyson series as (set ):

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Written out in full in momentum space, a Feynman propagator takes the form:

Using this form we can now write the matrix element in full:

It turns out that all of these integrals out the front simply act to cancel off all the external ‘legs’ of the Feynman

diagrams. Thus, in the end we get:

Using:

We thus have at last an expression for the term :

We can summarise the set of rules that allow us to write these M factors down directly in the momentum-space

Feynman rules:

1. Internal lines get propagators

2. Vertices represent one interaction each, and get a factor of each

3. External lines do not get any propagators

4. Integrate over all undetermined (internal) 4-momenta

5. Sum over all possible diagrams

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Key steps in getting Feynman rules LSZ Reduction Formula:

Dyson Series Expansion:

Wick’s Theorem:

Feynman Propagator:

Feynman rules for scalar field theory Propagators get:

Vertices get:

External lines get nothing.

For derivative terms in Lagrangian, all particles exiting a vertex get factors of for outgoing particle, while all

particles going into a vertex get a factor of for incoming particles.

Momentum is conserved at each vertex.

Integrate over all undetermined internal loop momenta.

Feynman rules for QED The QED Lagrangian is:

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Propagators are:

External particles get polarisation vectors or spinors:

Interaction sites get a factor of gamma:

Propagators go between spinors, with spinors in opposite order to particle flow (barred to left):

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Integrate over internal loops and sum over spin states (which is done by just taking the trace):

Extra minus signs for fermions:

Sum over final spins and average over initial spins (so factor of

should be introduced for two fermions):

Sum over final photon polarisations:

Using the Ward identity for sum of all diagrams only, this is even simpler:

Get by taking adjoint:

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(Also need to change and indexs to new letter)

Relativistic kinematics

In the massless case:

Mandelstam variables

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Renormalisation

We want to compute corrections to the Coulomb potential

. By the Ward identity these will take the form:

Where we have switched to Fourier space and is the loop integral from the Feynman diagram.

We use Feynman’s integral trick, Wick rotation, dimensional regularisation, and converting to 4d spherical

coordinates to write this integral in a form that localises the divergent part to a single term:

Now to use this result, we need to experimentally measure the constant at some momentum scale . This will

yield a finite result, and is called the renormalisation condition. Thus we have:

Equating prediction and experiment we get:

At this point both sides of this equation are formally infinite, because and are both divergent. However we can

substitute this back into our original formula for the potential to get:

We only need to keep the second order correction terms so up to , leaving:

This equation therefore gives us the second-order correction to the Coulomb potential in terms of the physically

measurable and the expression . All terms in this equation are now finite!

Mass renormalisation follows much the same process, except here we add counter-terms to the Lagrangian in order

to cancel off the divergent part of the integrals. This is done in accordance with some subtraction scheme which

specifies how exactly we determine these counter-terms.

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Gamma matrix identities Defining property of gamma matrices:

Basic identities:

p/

p/ p/

Trace identities:

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Feynman diagrams Process Equation Diagrams

Rutherford scattering

Compton scattering

Moller scattering

Bhabha scattering

Electron-positron annihilation

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Other useful relations

and

For on-shell particles ; for on-shell relativistic particles

Integration by parts:

When expanding out a p/ term you must introduce all new indices

∂/ p/

CM frame: , lab frame:

In the Weyl basis, and

is called the probability amplitude

Use same spin and particle labels across diagrams of the same process

Can only drop terms if relevant momenta are not on-shell (need to check this)

Total momentum going into a node must equal total momentum going out – ignore particle arrows

k/ p/ ∂/

s and s all commute with s and s, but not with each other

By symmetry and diagrams have the some , just swap variables

CM cross section:

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Exercises Exercise 7

Exercise 8

Exercise 11