Quantum Field Theory

There are no definite standard for quantum field theory. These are the notation I will use in this document.

• \( \mathbf{x} \) spacial position
• \( x \) four-position
• \( \mathbf{k} \) spacial wave vector
• \( k \) four-wave vector

Let us also suppress \( \hbar \) during the calculation, since they can always be recovered by dimensional analysis.

• \( \ket{\phi} \)

A quantum state with infinite degree of freedom, one for each point in space. Note that time is not considered here.

We will even remove the ◌̂ for conciseness, whenever it is clear.

The Klein-Gordon Lagrangian for a complex scalar (spin-0) field \( \phi \) is \[ \mathcal{L} = \partial^{\mu}\phi \partial_{\mu}\phi^* - m^2\phi^* \phi \] in mostly positive metric.

This is the simplest Lagrangian that is

• local: only in terms of local quantities such as \( \phi, \partial_{\mu}\phi \),
• first order derivative in time (from experience),
• Lorentz invariant.

Using the natural units, the classical equation of motion can be written as: \[ (\partial^{\mu}\partial_{\mu} + m^2)\phi = 0. \]

The solution can be easily found using the correpondence between the algebra of classical quantities and the algebra of observables within linear order: \[ \phi(x) = \int \frac{\dd[3]{\mathbf{k}}}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{k}}}} \left(a(\mathbf{k}) e^{-i\omega_{\mathbf{k}}t + i\mathbf{k}\vdot \mathbf{x}} + a^{\dagger}(\mathbf{k}) e^{i\omega_{\mathbf{k}}t - i\mathbf{k}\vdot \mathbf{x}}\right). \] This operator will detect and evolve each \( \mathbf{k} \) eigenstate of the initial state, and find the field variable at \( x \).

Fock space is a finite sum of Hilbert space \( \mathcal{H}_n \) of \( n \) particle systems: \[ \mathcal{H} := \ket{0} \oplus \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \cdots \oplus \mathcal{H}_m. \]

In non-relativistic quantum mechanics, the propagator in Heisenberg picture is given by: \[ G(x, x') := \braket{\mathbf{x}, t}{\mathbf{x}', t'} \] where \( \ket{\mathbf{x}, t} \) is the eigenvector of the position operator \( \hat{\mathbf{x}}(t)\) at time \( t \) with eigenvalue of \( \mathbf{x} \). So, \( \ket{\mathbf{x}, t} \) is the state in which after time \( t \), the amplitude converges at position \( \mathbf{x} \).

In quantum field theory, there is no position operator. We use the analog of the position eigenstate \[ \ket{x} := \phi(x)\ket{0}, \] and the propagator can be given in various ways:

where \( \theta \) is the unit step function. \( G_F \) is called the Feynman green function, or time-ordered correlation function, \( G_R \) the retarded function, \( G_A \) the advanced function.

Due to the spacetime translation symmetry and Lorentz invariance of the vacuum state, \( G \) can be fully described by a function of single variable \( \tilde{G}( (x-x')^2) \): the spacetime distance.

For the real scalar field, \( G \) can be written explicitly by substituting \( \phi(x) \) in:

Since the Fourier transform is defined in the same form: \[ G(k) := \int \dd[4]{x} G(x) e^{-ik\cdot x},\quad G(x - x') = \int \frac{\dd[4]{k}}{(2\pi)^4} G(k) e^{ik\cdot (x - x')}, \] we find that \[ G_+(k) = 2\pi\, \theta(k^0)\, \delta(k^2 + m^2). \]

From the Fourier transform of the operator equation (with mostly minus convention) \[ (-\partial_{\mu}\partial^{\mu} + m^2)G(x, x') = i\delta^{(4)}(x-x') \] where partial derivative is with respect to \( x \), we get: \[ (k^2 + m^2)G(k) = -i \implies G(k) = \frac{-i}{k^2 + m^2}. \] When we try to recover the original function, we find singularity at \( \omega = \pm\omega_{\mathbf{k}} \). This is resolved by going around the poles in the complex domain with a contour integral. Depending on the choice of detour, upper or lower half plane, we get different propagator. When upper, upper is chosen, we get \( G_R \), for lower, lower we get \( G_A \), and lower, upper we get \( G_F \).

This propagator gives complete information about the n-point correlation function: \[ G(x_1, x_2, \dots, x_n) := \ev{ \mathrm{T}\phi(x_1)\phi(x_2)\cdots \phi(x_n)}{0}. \] where \( \mathrm{T} \) is the time-ordering that orders the operators next to it with respect to time.

The n-point function can be expressed in terms of sums of products of correlation function.

In one dimensional case, the transition amplitude \[ \mel{x_b}{e^{-i\hat{H}t/\hbar}}{x_a} \] is generally not easy to calculate, but when we consider each small timestep \( \Delta t \) separately,

in each step the exponential is separated with the help of Baker-Campbell-Hausdorff formula: \[ \mel{x_{i+1}}{e^{-i \Delta t ( \hat{p}^2/2m + V(\hat{x}) )}}{x_i} = \mel{x_{i+1}}{e^{-i \Delta t \hat{p}^2/2m} e^{-i\Delta t V(\hat{x}) }}{x_i} + O(\Delta t^2). \] This can be further simplified to: \[ \mel{x_{i+1}}{e^{-i \Delta t \hat{p}^2/2m} e^{-i\Delta t V(\hat{x}) }}{x_i} \stackrel{\Delta t\to 0}{\to} \sqrt{\frac{m}{2\pi \hbar i \Delta t}} e^{i\Delta t L(x, \dot{x})|_{t_i}/\hbar}. \]

The resulting integral is called the path integral: \[ \int_{x(t_a) = x_a}^{x(t_b) = x_b} \mathcal{D}x\, e^{i\int_{t_i}^{t_f} \dd{t} L(x, \dot{x})} :=\lim_{n\to \infty} \left( \frac{m}{2\pi \hbar i \Delta t} \right)^{\frac{n}{2}} \int \dd{x_1}\dd{x_2}\cdots \dd{x_{n-1}} \exp \left( i\Delta t L \left(x_{n-1}, \frac{x_b - x_{n-1}}{\Delta t} \right) \big/\hbar\right)\cdots \exp \left( i\Delta t L \left(x_i, \frac{x_{i+1} - x_i}{\Delta t} \right) \big/\hbar\right)\cdots \exp \left( i\Delta t L \left(x_a, \frac{x_{1} - x_a}{\Delta t} \right) \big/\hbar\right). \] This integral can be reconceptualized into integrating over all possible path \( x(t) \) that strats \( x_a \) and ends at \( x_b \), and more succictly written as: \[ \int_{x_a}^{x_b} \mathcal{D}x\, e^{iS[x]}. \]

For a system with more degrees of freedom, we simply have to account for every degree of freedom at each timestep. A quantum field has uncountably infinite degrees of freedom but it is not much different conceptually: \[ \int_{\phi(\mathbf{x}, t_a) = \phi_a(\mathbf{x})}^{\phi(\mathbf{x}, t_b) = \phi_b(\mathbf{x})} \mathcal{D} \phi\, e^{i \int_{t_a}^{t_b} L(\phi, \dot{\phi})} = \int_{\phi_a}^{\phi_b} \mathcal{D}\phi\, e^{iS[\phi]}. \] Notice each path is a evolution of field \( \phi(\mathbf{x}, t) \) from configuration \( \phi_a \) to \( \phi_b \).

• Generating Functional

We introduce the source term into the path integral: \[ Z[J] := \int \mathcal{D}\phi\, e^{i\int \dd[4]{x} (\mathcal{L} + J\phi)} \] where \( J \) is a function of spacetime \( J(x) \) called source function.

With the definition of functional derivative: \[ \frac{\delta J(t_b)}{\delta J(t_a)} := \delta(t_b - t_a) \] we can express the n-point correlation function in terms of the derivative of \( Z[J] \): \[ G(x_1, x_2, \dots, x_n) = \frac{1}{i^nZ_0} \frac{\delta^n Z[J]}{\delta J(x_1) \delta J(x_2) \cdots \delta J(x_n)} \] where \( Z_0 := Z[0] \).

This derivative treats each value of \( J \) at a spacetime coordinate \( x_i \) as an independent variable.

\( Z[J] \) is called the generating functional, because the coefficients of its power series expansion are the correlation functions: \[ Z[J] = Z_0 \left[ i\sum_{x_1} \frac{1}{i Z_0}\frac{\delta Z[J]}{\delta J(x_1)}J(x_1) + i^2 \sum_{x_1, x_2} \frac{1}{i^2 Z_0}\frac{\delta^2 Z[J]}{\delta J(x_1) \delta J(x_2)} J(x_1) J(x_2) + \cdots \right] \]

There are two kinds of divergence in quantum field theory: ultraviolet and infrared.

The infrared divergence is easy to deal with. It happens due to the fact that the theory is assuming infinite space which is not realistic. The contribution from extreme distances adds up and produce infinity. Regularization of dividing out the volumn removes this infinity.

Ultraviolet divergence happens due to the failure of the theory in modeling the reality. In particular, the terms involving unrealistic extreme energies eventually add up and produce infinity.

\[ \Gamma \]

• MIT 8.323 Relativistic Quantum Field Theory I, Spring 2023 - YouTube
• Spinors for Beginners 21: Quantum Field Theory from the ground up - YouTube
• Dirac equation - Wikipedia
• Deriving the Dirac Equation - YouTube
• Spinors for Beginners - YouTube
• I Never Understood How Quantum Particles Tunnel…Until now! - YouTube