Index, Roadmap, and Conventions

These notes are a self-contained, graduate-level introduction to relativistic quantum field theory (QFT), written inspired by notes taken in Prof. Polyakov’s lectures and the book of Peskin–Schroeder: start from canonical quantization, develop perturbation theory and diagrammatics, introduce the path integral and generating functionals, and then build toward renormalization and gauge theories.

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Roadmap

One-semester intro QFT:

1. Quantum harmonic oscillator and Fock space
   Why it matters: every Fourier mode of a free field is an oscillator.

2. Free scalar fields
   Klein–Gordon equation, canonical quantization, propagators.

3. Interaction picture and Wick’s theorem
   Dyson series, time ordering, contraction rules, diagrammatics.

4. Path integrals
   Gaussian functional integrals, generating functionals, connected/1PI, effective action.

5. Renormalization and the renormalization group
   Power counting, counterterms, running couplings, physical interpretation.

6. Spin and fermions
   Lorentz group, spinor representations, Dirac fields, Grassmann path integrals.

7. Gauge fields and QED basics
   Gauge invariance, covariant derivatives, Ward identities (at least conceptually).

A core structural idea is:

Additive spectrum: for a free theory, energies add across independent modes, and multi-particle states are naturally described by a Fock space.

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Physical units

We use natural units:

$$\hbar = c = 1,$$

and later (for thermal field theory) also $k_B = 1$ unless explicitly stated.

Dimensional analysis is always performed in units of energy (mass).

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Spacetime and index conventions

We use the mostly-minus metric:

$$\eta_{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1), \qquad x^\mu = (t,\mathbf{x}), \qquad p^\mu = (p^0,\mathbf{p}) = (E,\mathbf{p}).$$

Dot products:

$$p\cdot x \equiv p_\mu x^\mu = p^0 t - \mathbf{p}\cdot \mathbf{x}.$$

Derivatives:

$$\partial_\mu = \frac{\partial}{\partial x^\mu} = (\partial_t,\nabla), \qquad \partial^\mu = \eta^{\mu\nu}\partial_\nu = (\partial_t,-\nabla).$$

The d’Alembertian:

$$\square \equiv \partial_\mu\partial^\mu = \partial_t^2 - \nabla^2.$$

Levi–Civita tensor:

$$\varepsilon^{0123}=+1,$$

and indices are raised/lowered with $\eta$.

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Fourier transform conventions

All factors of $2\pi$ are kept with the momentum-space integration measure. We adopt:

$$f(x) \;=\; \int \frac{d^4p}{(2\pi)^4}\, e^{-ip\cdot x}\,\tilde f(p), \qquad \tilde f(p) \;=\; \int d^4x\, e^{+ip\cdot x}\,f(x).$$

In 3D:

$$g(\mathbf{x})=\int \frac{d^3p}{(2\pi)^3}\, e^{i\mathbf{p}\cdot\mathbf{x}}\,\tilde g(\mathbf{p}), \qquad \tilde g(\mathbf{p})=\int d^3x\, e^{-i\mathbf{p}\cdot\mathbf{x}}\, g(\mathbf{x}).$$

(There are other sign conventions in the literature; what matters is to be consistent.)

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Delta functions

With the above Fourier conventions,

$$\delta^{(4)}(x-y) = \int \frac{d^4p}{(2\pi)^4}\, e^{-ip\cdot(x-y)}, \qquad (2\pi)^4\delta^{(4)}(p-q)=\int d^4x\, e^{i(p-q)\cdot x}.$$

Similarly,

$$\delta^{(3)}(\mathbf{x}-\mathbf{y}) = \int \frac{d^3p}{(2\pi)^3}\, e^{i\mathbf{p}\cdot(\mathbf{x}-\mathbf{y})}.$$

We also use the step function $\theta(t)$, with $\theta(t>0)=1$ and $\theta(t<0)=0$.

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Commutators and adjoints

Commutator and anticommutator:

$$[A,B]\equiv AB-BA, \qquad \{A,B\}\equiv AB+BA.$$

Hermitian conjugation:

$$(\lambda A)^\dagger = \lambda^* A^\dagger, \qquad (AB)^\dagger = B^\dagger A^\dagger.$$

Inner products follow Dirac notation: $\langle A|B\rangle$ is linear in $|B\rangle$ and anti-linear in $\langle A|$.

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Momentum eigenstates and normalization

There are two common normalizations for one-particle momentum eigenstates. We will use both, but we will always state which one is being used.

1) “Delta-function normalization” (common in canonical quantization)

$$\langle \mathbf{p}|\mathbf{q}\rangle = (2\pi)^3 \delta^{(3)}(\mathbf{p}-\mathbf{q}), \qquad \int \frac{d^3p}{(2\pi)^3}\,|\mathbf{p}\rangle\langle\mathbf{p}| = \mathbf{1}.$$

This is convenient when creation/annihilation operators satisfy

$$[a_{\mathbf{p}},a^\dagger_{\mathbf{q}}]=(2\pi)^3\delta^{(3)}(\mathbf{p}-\mathbf{q}).$$

2) Lorentz-invariant normalization (common in scattering theory)

Define Lorentz-invariant one-particle states $|\mathbf{p}\rangle_{\rm LI}$ so that

$${}_{\rm LI}\langle \mathbf{p}|\mathbf{q}\rangle_{\rm LI} = (2\pi)^3\,2E_{\mathbf{p}}\,\delta^{(3)}(\mathbf{p}-\mathbf{q}), \qquad E_{\mathbf{p}}=\sqrt{\mathbf{p}^2+m^2},$$

and completeness becomes

$$\int \frac{d^3p}{(2\pi)^3\,2E_{\mathbf{p}}}\, |\mathbf{p}\rangle_{\rm LI}\,{}_{\rm LI}\langle \mathbf{p}| = \mathbf{1}.$$

The measure

$$\frac{d^3p}{(2\pi)^3\,2E_{\mathbf{p}}}$$

is Lorentz invariant (a key reason it appears throughout relativistic QFT).

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Why “additive spectrum” is the first clue of QFT

• For decoupled subsystems, the total energy is the sum of energies.
• In QFT, a free field is an infinite set of decoupled oscillators labeled by momentum $\mathbf{p}$.
• Therefore multi-particle states look like “occupation numbers” $n_{\mathbf{p}}$ for each momentum mode.

We will build this carefully, starting from a single harmonic oscillator in the next chapter.

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Exercises

Below are short “conventions drills.” They are worth doing once; later calculations become much less error-prone.

Exercise 1: Delta function from Fourier transform

Show that with the convention

$$f(x)=\int \frac{d^4p}{(2\pi)^4}e^{-ip\cdot x}\tilde f(p),$$

one has

$$\delta^{(4)}(x-y)=\int\frac{d^4p}{(2\pi)^4}\,e^{-ip\cdot(x-y)}.$$

Solution

Start from the inversion formula:

$$\tilde f(p)=\int d^4x\,e^{ip\cdot x}f(x).$$

Plug into the forward transform:

$$f(x)=\int\frac{d^4p}{(2\pi)^4}e^{-ip\cdot x}\int d^4y\,e^{ip\cdot y}f(y).$$

Swap integrals:

$$f(x)=\int d^4y\,\Bigl[\int\frac{d^4p}{(2\pi)^4}e^{-ip\cdot(x-y)}\Bigr]\,f(y).$$

Since this holds for arbitrary test functions $f(y)$, the bracket must be $\delta^{(4)}(x-y)$.

Exercise 2: Dispersion relation with $(+,-,-,-)$

Show that the mass-shell condition $p^2=m^2$ (with $p^2=p_\mu p^\mu$) implies

$$E_{\mathbf{p}}=\sqrt{\mathbf{p}^2+m^2}.$$

Solution

With $\eta=(+,-,-,-)$,

$$p^2 = (p^0)^2 - \mathbf{p}^2 = E^2-\mathbf{p}^2.$$

Setting $p^2=m^2$ gives

$$E^2-\mathbf{p}^2 = m^2 \quad\Rightarrow\quad E^2 = \mathbf{p}^2+m^2.$$

Taking the positive-energy branch for particles yields $E_{\mathbf{p}}=\sqrt{\mathbf{p}^2+m^2}$.

Exercise 3: Lorentz-invariant measure (boost along one axis)

Consider a boost in the $z$-direction. Show (sketch) that

$$\frac{d^3p}{2E_{\mathbf{p}}}$$

is invariant under such boosts.

Solution (sketch)

A standard derivation starts from the manifestly invariant object

$$d^4p\,\delta(p^2-m^2)\,\theta(p^0).$$

The delta function restricts to the positive-energy mass shell:

$$\delta(p^2-m^2)\,\theta(p^0) = \frac{1}{2E_{\mathbf{p}}}\delta(p^0-E_{\mathbf{p}}).$$

Integrating over $p^0$ gives

$$\int d^4p\,\delta(p^2-m^2)\,\theta(p^0)\,(\cdots) = \int \frac{d^3p}{2E_{\mathbf{p}}}\,(\cdots)\bigg|_{p^0=E_{\mathbf{p}}}.$$

Since the left-hand side is Lorentz invariant, the measure $d^3p/(2E_{\mathbf{p}})$ must be invariant.