Fernando Garcia
Home Research Blog Other About me
Quantum Field Theory
Formulas
Sources
The main influences for this collection include Mark Srednicki's "Quantum Field Theory" and Peskin and Schroeder's "An introduction to Quantum Field Theory."
Contents
Natural units, dimensional analysis.
Mathematical facts and identities relevant to QFT.
Scalar fields and canonical quantization.
Lehmann–Symanzik–Zimmermann (LSZ) reduction formula.
Path Integrals I - Quantum Mechanics.
Path Integrals II - (Scalar) Free field theory.
Path Integrals III - (Scalar) Interacting field theories.
Natural units, dimensional analysis.
\[\hbar =c=1 \]Denoting time by $T $, length by $L $, and mass by $M$, we see that
\begin{align*} T &= c^{-1} L \\ L &= \hbar c M^{-1} \end{align*}Everything can be expressed in powers of mass. We denote its dimension by such a power:
\begin{align*} \left[ m \right] &= +1 \\ \left[ x^{\mu } \right] &= -1 \\ \left[ \partial ^{\mu } \right] &= +1 \\ \left[ d^dx \right] &= -d \end{align*}The action and Lagrangian (density) are such that
\begin{align*} \left[ S \right] &= 0 \\ \left[ \mathcal{L} \right] &= d \end{align*}For $\phi $ a scalar field, we have:
\begin{equation} \left[ \phi \right] =\frac{1}{2}(d-2) \end{equation}If the natural unit of mass is the electron volt, then length has units of $1/\text{eV} $, time has units of $1/\text{eV} $, energy has units of $\text{eV} $, potential has units of $\text{eV} $, charge is dimensionless.
\begin{align*} h &= 2\pi \\ \epsilon _0 &= \frac{1}{4\pi }\\ \mu _0 &= 4\pi \end{align*}Mathematical facts, notation, and identities relevant to QFT.
Notation:
\[ f\stackrel{\leftrightarrow}{\partial _{\mu }}g =f(\partial _{\mu }g)-(\partial _{\mu }f)g\]Identities:
\[\int d^3 xe^{i\mathbf{q}\cdot \mathbf{x}}=(2\pi )^3 \delta ^3 (\mathbf{q}) \]Scalar fields and canonical quantization.
Using natural units $\hbar =c=1 $...
Klein-Gordon equation:
\[ \left( -\partial ^2 +m^2 \right) \phi (x)=0 \]Plane wave solution to the KG Equation:
\[\phi (x)=\exp \left( i\mathbf{k}\cdot \mathbf{x}\pm i\omega t \right) ,\;\;\;\omega =+\sqrt{\mathbf{k}^2 +m^2 } \]General solution to the KG Equation (where $a(\mathbf{k}) $ is an arbitrary function):
\[\phi (x)=\int \frac{d^3 k}{(2\pi )^3 2\omega } \left[ a(\mathbf{k})e^{ikx}+a^* (\mathbf{k})e^{-ikx} \right] \]Scalar field theory Lagrangian (density), with $\Omega _0 $ a constant:
\[\mathcal{L}=-\frac{1}{2}\partial ^{\mu }\phi \partial _{\mu }\phi -\frac{1}{2}m^2 \phi ^2 +\Omega _0 \]Definition of the Hamiltonian density:
\[\mathcal{H}=\Pi \dot{\phi }-\mathcal{L} \]Where the conjugate momenta is (for a scalar field theory):
\begin{align*} \Pi (x) &= \frac{\partial \mathcal{L}}{\partial \dot{\phi }} \\ &= \dot{\phi }(x) \end{align*}The Hamiltonian (density) for a free scalar theory:
\[\mathcal{H}=\frac{1}{2}\Pi ^2 +\frac{1}{2}(\nabla \phi )^2 +\frac{1}{2}m^2 \phi ^2 -\Omega _0 \]Canonical commutation relations (for the field operators)
\begin{align*} \left[ \phi (\mathbf{x},t),\phi (\mathbf{y},t) \right] &= 0 \\ \left[ \Pi (\mathbf{x},t),\Pi (\mathbf{y},t) \right] &= 0 \\ \left[ \phi (\mathbf{x},t),\Pi (\mathbf{y},t) \right] &= i\delta ^3 (\mathbf{x}-\mathbf{y}) \end{align*}Canonical commutation relations (for the creation and annihilation operators)
\begin{align*} \left[ a(\mathbf{k}),a(\mathbf{k}') \right] &= 0 \\ \left[ a^{\dagger}(\mathbf{k}),a^{\dagger}(\mathbf{k}') \right] &= 0 \\ \left[ a(\mathbf{k}),a^{\dagger}(\mathbf{k}') \right] &= (2\pi )^3 2\omega \delta ^3 (\mathbf{k}-\mathbf{k}') \end{align*}The free scalar Hamiltonian is:
\[H=\int \frac{d^3 k}{2(2\pi )^3 }a^{\dagger}(\mathbf{k})a(\mathbf{k})+(\mathcal{E}_0 -\Omega _0 )V \]Where we have the total zero-point energy density of all the oscillators.
\[\mathcal{E}_0 =\frac{1}{2}(2\pi )^{-3} \int d^3 k\omega \]The concluding remarks are relevant:
(1) When studying the free theory, one sets $\Omega _0 =\mathcal{E}_0 $ to avoid performing an ultraviolet cutoff at this point.
(2) Further, the differential
\[\tilde{dk}=\frac{d^3 k}{(2\pi )^3 2\omega } \]Is frequently used and known as the Lorentz-invariant differential.
Lehmann–Symanzik–Zimmermann (LSZ) reduction formula
In the case of $n $ incoming and $n' $ outgoing particles, the Lehmann-Symanzik-Zimermann reduction formula (LSZ) is:
\[\langle f|i \rangle =i^{n+n'}\int d^4 x_1 e^{ik_1 x_1 }(-\partial ^2 _{1}+m^2 )\cdots d^4 x'_{1}e^{-ik'_1 x'_1 }(-\partial _{1'}^2 +m^2 )\cdots \langle 0|T\phi (x_1 )\cdots \phi (x'_1 )\cdots |0 \rangle \]Where $T $ is a time-ordering symbol, and $\langle f|i \rangle $ denotes the scattering amplitude.
The derivation of this formula is usually done before talking about interacting fields in-depth. As such, it is relevant to note the necessary conditions/assumptions:
(1) The ground state $| 0 \rangle $ is unique, and it is such that it possesses no energy and momentum.
(2) The first excited state is a 1-particle state, the second is a 2-particle state (mass $m $, $2m $, etc.). For a $n $-particle state, the lowest possible energy (assuming no bound states) is $nm $ (that is, $\mathbf{k}=0 $)
(3) The fields must obey
\begin{align*} \langle 0|\phi (0)|0 \rangle &= 0 \\ \langle p|\phi (x)|0 \rangle &= e^{-ikx} \end{align*}Path Integrals I - Quantum Mechanics
In the Heisenberg picture, the transition amplitude for the event/transition:
Is denoted by $\langle q''|t''|q',t' \rangle $.
It can be shown that
\[\langle q'',t''|q',t' \rangle =\int \mathcal{D}q \exp \left( i\int_{t'}^{t''} dt\;L(\dot{q}(t),q(t)) \right) \]Where $L $ is the Lagrangian of the system. This is one of the most remarkable points of the path integral approach to Quantum Mechanics (and QFT): it allows us to move from the Hamiltonian to the Lagrangian.
The above $\mathcal{D}q $ integration is over all paths such that $q(t')=q' $ and $q(t'')=q'' $.
For additional operators:
\begin{align*} \langle q'',t''|Q(t_1 )|q',t' \rangle &=\int \mathcal{D}p\mathcal{D}q\;q(t_1 )e^{iS} \\ \int \mathcal{D}p\mathcal{D}q\;q(t_1 )q(t_2 )e^{iS} &= \langle q'',t''|TQ(t_1 )Q(t_2 )|q',t' \rangle \end{align*}We define the functional derivative as:
\[\frac{\delta }{\delta f(t_1 )}f(t_2 )=\delta (t_1 -t_2 ) \](product, chain, etc. rules hold).
By considering the modified (added external forces) Hamiltonian:
\[H(p,q)\rightarrow H(p,q)-f(t)q(t)-h(t)p(t) \]We define
\[\langle q'',t''|q',t' \rangle _{f,h}=\int \mathcal{D}p\mathcal{D}q \exp \left( i \int_{t'}^{t''} dt \left( p\dot{q}-H+fq+hp \right) \right) \]So that
\[\langle q'',t''|TQ(t_1 )\cdots P(t_n )\cdots |q',t' \rangle =\frac{1}{i}\frac{\delta }{\delta f(t_1 )}\cdots \frac{1}{i}\frac{\delta }{\delta h(t_n )}\cdots \langle q'',t'' | q',t' \rangle _{f,h}\Big|_{f=h=0} \]Whenever the initial and final states are the ground state:
\[\langle 0|0 \rangle _{f,h}=\int \mathcal{D}p\mathcal{D}q \;\exp \left( i \int_{-\infty }^{\infty } dt \left( p\dot{q}-(1-i\epsilon )H+fq+hp \right) \right) \]Where we can let $h=0 $:
\[\langle 0|0 \rangle _{f}=\int \mathcal{D}p\mathcal{D}q \;\exp \left( i \int_{-\infty }^{\infty } dt \left( p\dot{q}-(1-i\epsilon )H+fq \right) \right) \]If we are interested well-behaved systems (interactions written in terms of $q $, no more than quadratic in $P $ without $Q $, and for time-ordered products of $Q $s) such as the harmonic oscillator:
\[H_{\text{H.O.} }(P,Q)=\frac{1}{2m}P^2 +\frac{1}{2}m\omega ^2 Q^2 \]For which
\begin{align*} \langle 0|0 \rangle _f &= \exp \left( \frac{i}{2}\int_{-\infty }^{\infty } \frac{dE}{2\pi }\frac{\tilde{f}(E)\tilde{f}(-E)}{-E^2 +\omega ^2 -i\epsilon } \right) \\ &= \exp \left( \frac{i}{2}\int_{-\infty }^{\infty } dtdt'\;f(t)G(t-t')f(t') \right) \end{align*}Where $G(t-t') $:
\[G(t-t')=\int_{-\infty }^{\infty } \frac{dE}{2\pi }\frac{e^{-iE(t-t')}}{-E^2 +\omega ^2 -i\epsilon } \]Is a Green's function:
\[ \left( \frac{\partial ^2 }{\partial t^2 }+\omega ^2 \right) G(t-t')=\delta (t-t') \]$G $ can be written as
\[G(t-t')=\frac{i}{2\omega }\exp \left( -i\omega |t-t'| \right) \]Which can be used in the general formula for the ground-state expectation value of some time-ordered product of many $Q(t) $s:
\[\langle 0|TQ(t_1 )\cdots Q(t_{2n})|0 \rangle =\frac{1}{i^n }\sum_{\text{pairings/contractions} }^{} G(t_{i_1 }-t_{i_2 })\cdots G(t_{i_{2n-1}}-t_{i_{2n}}) \]Where it is relevant to note that if the number of $Q(t) $s in the product is odd, then the expectation value vanishes.
Path Integrals II - (Scalar) Free field theory
The path integral in a free-field scalar theory is (as a generalization of the above):
\[Z_0(J)\equiv \langle 0|0 \rangle _J=\int \mathcal{D}\phi \;\exp \left( i \int d^4 x \left[ \mathcal{L}_0 +J\phi \right] \right) \]Where $q $ was replaced by the classical field $\phi (\mathbf{x},t) $, $Q $ by the operator corresponding to $\phi (\mathbf{x},t) $, and $f $ became a classical source $J(\mathbf{x},t) $.
This path integral goes over all paths in the space of field configurations. The measure $\mathcal{D}\phi $ is proportional to $\prod _{x}d\phi (x) $.
Remark: If we fourier transform the field $\phi $:
\begin{align*} \phi (x) &= \int \frac{d^4 k}{(2\pi )^4 }e^{ikx}\tilde{\phi }(k) \\ \tilde{\phi }(k) &= \int d^4 x e^{-ikx}\phi (x) \end{align*}The action $S_0 =\int d^4 x \left[ \mathcal{L}_0 +J\phi \right] $ becomes one that resembles that of the harmonic oscillator. By introducing the Feynman propagator:
\[\Delta (x-x')=\int \frac{d^4 k}{(2\pi )^4 }\frac{e^{ik(x-x')}}{k^2 +m^2 -i\epsilon } \]We are able to write $Z_0 $ as:
\[Z_0 (J)=\exp \left( \frac{i}{2}\int d^4 xd^4 x'\;J(x)\Delta (x-x')J(x') \right) \]So
\[\langle 0|T\phi (x_1 )\cdots |0 \rangle =\frac{1}{i}\frac{\delta }{\delta J(x_1 )}\cdots Z_0 (J)\Big|_{J=0} \]For example:
\[\langle 0|T\phi (x_1 )\phi (x_2 )|0 \rangle =\frac{1}{i}\Delta (x_2 -x_1 ) \]In general (for even number of $\phi $s in the time ordered product, otherwise it vanishes), we have Wick's theorem:
\[\langle 0|T\phi (x_1 )\cdots \phi (x_{2n})|0 \rangle =\frac{1}{i^n }\sum_{\text{pairings/contractions} }^{} \Delta (x_{i_1 }-x_{i_2 })\cdots \Delta (x_{i_{2n-1}}-x_{i_{2n}}) \]