# Quantum Field Theory

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## Under construction¶

Note

These notes on quantum field theory are under construction.

QFT is simply quantum physics where the state of a system is a field. This raises a swarm of difficulties, but also a host of important and unifying physical ideas, between quantum physics, relativity and statistical mechanics. It is the language of the (experimentally validated) fundamental theories of physics.

Resources

*Quantum Field Theory in a Nutshell*- David Tong's notes
- David Skinner's notes
- These notes: https://nicf.net/articles/qft-free-fields/

## Dimensions¶

Working in natural units, we have \(\hbar = c = 1\). Since they are dimensionless, we get:

and

so \(T = L = M^{-1} = E^{-1}\)

So large energy scale (UV) means small length scale, and small energy (IR) means large length.

## Basic derivations with path integrals¶

Note

Refer to the quantum notes for how path integrals work.

Suppose we have a Lagrangian:

so that \(\mathcal{L}(\phi(x),\partial \phi(x), J(x)) = \frac{1}{2}((\partial\phi)^2(x) -m^2\phi^2(x) )+J(x)\phi(x)\) is what we integrate over, i.e.:

Here \(D\phi\) indicates an integral ranging over all functions \(\phi\) with boundaries \(\phi(0)= q_I\) and \(\phi(T)=q_F\). The integral in the exponent ranges over all space, and time \([0,T]\).

\(J\) is understood as an excitation of the field \(\phi\).

Basic properties of Gaussian integrals allow us to derive:

with

### Force¶

Suppose \(J(\overrightarrow x, t) = \delta^3(x-x_1) + \delta^3(x-x_2):= J_1(x) + J_2(x)\).

Taking \(|q_I\rangle=|q_F\rangle = |0\rangle\), so that \(e^{-W(J)} = \langle 0 | e^{-iHT} |0 \rangle = e^{-iE_0T}\), where \(E_0\) is the ground state energy of the system in question, a simple calculation shows that:

## Calculation

Ignore the \(J_iJ_i\) terms, and consider the \(J_1J_2\) term. Plugging in the definition of \(D\), we have:

so that with \(iW = -iE_0T\), recalling that \(dx^0\) ranges over \([0,T]\), observing that \(\overrightarrow k^2 + m^2\) is positive so that \(\epsilon\) can be taken to \(0\), and counting both the \(J_1J_2\) and \(J_2J_1\) terms, we obtain:

**Under construction**

Observing that with \(J=0\), i.e. in the absence of imposed excitations, \(E_0=0\), we see that the energy is lowered.

"Physically", we think of the two delta functions as sources/sinks of a particle, so that the exchange of a particle produces an attractive force.

## Hamiltonian approach (canonical quantization)¶

Recalling

we calculate the energy density \(T^0_0 = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi)}\frac{\partial \phi}{\partial x_0} - \mathcal{L} = \frac{1}{2}(\partial_0\phi\partial_0\phi+(\partial_i\phi\partial^i\phi)+m^2\phi^2)=\frac{1}{2}(\pi^2 + (\nabla\phi)^2 + m^2\phi^2)\)

### Example with Klein-Gordon Lagrangian¶

We start with a typical classical free theory, e.g. the scalar case:

Quantizing the derivation of the Klein-Gordon equation and its solution from the notes on mechanics, we have:

(and similarly for \(\pi(x)\)), with:

Omitting hats going forward, we have a 4-momentum operator:

so that:

We then define \(|0\rangle\) by the property \(\forall p, a_p|0\rangle = 0\), which exists since \(H\) is bounded below.

Eigenvectors of \(P^\mu\) are generated by

which is an eigenstate of \(H\) with \(\omega_p\) and of \(P\) with \(p\). **We consider such states particles**, noting their energy momentum relationship.

Similarly in position space:

## Interacting theory¶

and with \(H_I := U_0(t)H_{int}U_0^\dagger(t)\), we can propagate a system through time with:

where \(T(A(t_1)B(t_2)) = A(t_1)B(t_2)\) if \(t_1 > t_2\), else \(B(t_2)A(t_1)\).

## Green's function¶

The Green's function of an operator is characterized (up to some proportionality constant) by

## Particles, field theoretically¶

The standard interpretation of particles in a quantum field works like this. We take the Klein-Gordon field, and put it in a box. In this case, eigenvectors are discrete, and we see that the system is a product of independent harmonic oscillators.

Using the standard Dirac argument, we represent states by the number of creation operators applied to the vacuum of \(Sym(\mathbb{C}^n)\), where \(n\) is the number of oscillators.

Physically, we say that a state with \(n\) creation operators applied is a state with \(n\) particles. For instance \(a_p^\dagger|0\rangle\) is a state with one particle with momentum \(p\).

The Hamiltonian is \(H = \sum \omega_p a_p^\dagger a_p\), where \(\omega_p=\sqrt{|p|^2 + m^2}\), so that such a state is an eigenvector with eigenvalue \(\sqrt{|p|^2 + m^2}\), so that its 4-momentum is \((\sqrt{|p|^2 + m^2}, \overrightarrow p)\) and its mass is \(m\).

The fact that even n-particle states (for \(n>1\)) are eigenstates means that they are time preserved and so there are no interactions, which is why we call this a free field theory.

## Particles, categorically¶

This is a more abstract and general view of the above.

Recall that a representation is a functor from a group (viewed as a one object category whose morphisms are the group elements) to a category of Hilbert spaces \(\mathit{HILB}\).

Given a representation \(F_1 : G \to \mathit{HILB}\), states of a physical system are rays in \(H := F_1(G)\).

The irreps of \(F_1\) are interpreted physically as particles, whose states therefore live in subspaces of \(H\), so that the whole system is determined by the state of all the particles.

The dynamics of the physical system is captured by a natural transformation \(F_1 \mapsto F_2\). Spelled out, this is precisely an intertwining map between representations, i.e. a function \(S\) such that \(S \circ F_1(g) = F_2(g) \circ S\). In other words, the dynamics are symmetry preserving: you can change viewpoint before or after the dynamics, and you get the same result.

We now invoke Schur's lemma to great effect, in two ways.

First by a corollary of Schur's lemma, which tells us that if \(U\) is a natural transformation \(F_1 \to F_1\), then the irreps are exactly the eigenspaces of \(U\). Since energy \(H\) generates \(U\), we can label particles by their energy.

Second, Schur's lemma tells us that the only natural transformations between irreps are \(1\) (i.e. the identity) or \(0\). This means that particles cannot simply change into other particles (without some other interaction).

This only applies to irreps, so there are certainly non-trivial natural transformations between tensor products of irreps, e.g. \(H_1 \otimes H_2 \to H_3\). Feynman diagrams denote precisely these natural transformations.

## S-matrix¶

Suppose we start in a free state, i.e. a tensor product of 1 particle states, and end in one too. The map (a natural transformation as above) which connects them is the s-matrix.

## Under construction¶

More to follow.