1.4 Notations and conventions

Quantum mechanics:

• • We work in natural units in which the (reduced) Planck constant is one, i.e. \(\hbar =1\).

• • We do not use hats or other decorations to denote operators. It will be clear from the context if a symbol such as \(A\) denotes an operator or a classical quantity.

• • The commutator between two operators \(A\) and \(B\) is defined as

  \(\seteqnumber{0}{1.}{6}\)

  \begin{flalign} [A,B]:= A\,B - B\,A\quad . \end{flalign}

• • The canonical commutation relations (CCRs) between position \(q_a\) and momentum \(p^a\) operators, where \(a=1,\dots , N\) is an index running over the degrees of freedom, are given by

  \(\seteqnumber{0}{1.}{7}\)

  \begin{flalign} \label {eqn:CCRmechanics} [q_a,q_b] = 0 = [p^a,p^b]\quad ,\qquad [q_a,p^b] =\ii \,\delta _a^b= - [p^b,q_a]\quad , \end{flalign} where \(\delta _a^b\) denotes the Kronecker delta, i.e. \(\delta _a^b =1\) for \(a=b\) and \(\delta _a^b = 0\) else.

• • States in a Hilbert space are denoted by Dirac’s ket notation \(\ket {\psi }\in \HH \) and inner products by the bra-ket notation \(\braket {\phi }{\psi }\). The matrix-element of an operator \(A\) between two states \(\ket {\psi },\ket {\phi }\in \HH \) will be denoted by

  \(\seteqnumber{0}{1.}{8}\)

  \begin{flalign} \expect {\phi }{A}{\psi }:= \braket {\phi }{A\psi }\quad . \end{flalign}

• • The adjoint of an operator \(A\) will be denoted by \(A^\dagger \). By definition, one has

  \(\seteqnumber{0}{1.}{9}\)

  \begin{flalign} \braket {A^\dagger \phi }{\psi } = \braket {\phi }{A \psi } = \expect {\phi }{A}{\psi }\quad , \end{flalign} for all states \(\ket {\psi },\ket {\phi }\in \HH \). In particular, one can read the matrix-element \(\expect {\phi }{A}{\psi }\) in two different but equivalent ways: 1.) The operator \(A\) acts from left to right on the ket-state \(\ket {\psi }\), which yields \(A\ket {\psi }=\ket {A \psi }\) and hence \(\braket {\phi }{A \psi }\). 2.) The operator \(A\) acts from right to left on the bra-state \(\bra {\phi }\), which requires taking the adjoint \(\bra {\phi } A = \bra {A^\dagger \phi }\) and yields \(\braket {A^\dagger \phi }{\psi }\).

• • We work mostly in the Heisenberg picture, where states are time-independent. The time evolution of an operator \(A\) is described by Heisenberg’s equation

  \(\seteqnumber{0}{1.}{10}\)

  \begin{flalign} \frac {\dd A(t)}{\dd t} = \ii \,[H,A(t)]\quad , \end{flalign} subject to the initial condition \(A(0)=A\), where \(H\) is the Hamiltonian operator of the system. The solution of this equation is

  \(\seteqnumber{0}{1.}{11}\)

  \begin{flalign} A(t) = e^{\ii t[H,-]}\,A = e^{\ii H t}\,A \,e^{-\ii H t}\quad . \end{flalign}

Special relativity:

• • We work in natural units in which the speed of light is one, i.e. \(c=1\).

• • Our conventions for the \(d\)-dimensional Minkowski spacetime \((\bbR ^d,\eta )\) are

  \(\seteqnumber{0}{1.}{12}\)

  \begin{flalign} x = \begin{pmatrix} x^0\\ x^1 \\ \vdots \\ x^{d-2}\\ x^{d-1} \end {pmatrix} = \begin{pmatrix} t\\ x^1 \\ \vdots \\ x^{d-2}\\ x^{d-1} \end {pmatrix}\in \bbR ^d \quad ,\qquad \eta = \begin{pmatrix} -1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0\\ 0 & 0 & \cdots & 0 & 1 \end {pmatrix}\in \mathrm {Mat}_{d\times d}^{}(\bbR )\quad , \end{flalign} i.e. we use the ‘mostly plus’ convention for the Minkowski metric. The dimension \(d\) is sometimes taken to be \(4\), i.e. one time dimension and three space dimensions as observed in nature, but we will also consider examples of QFTs in other dimensions.

• • We will abbreviate the collection of space coordinates by boldface letters, i.e. we write

  \(\seteqnumber{0}{1.}{13}\)

  \begin{flalign} x = \begin{pmatrix} t\\ \mathbf {x} \end {pmatrix} := \begin{pmatrix} t\\ x^1 \\ \vdots \\ x^{d-2}\\ x^{d-1} \end {pmatrix}\in \bbR ^d\quad . \end{flalign}

• • We will often use index notation and write \(x^\mu \) for the spacetime coordinates and \(\eta _{\mu \nu }\) for the Minkowski metric. Greek letters \(\mu ,\nu ,\rho ,\dots \) from the middle of the alphabet will run over the whole range \(0,1,\dots ,d-1\) and, by Einstein’s summation convention, summations over repeated upper and lower indices will be suppressed. For the space coordinates \(\mathbf {x}\), we use the index notation \(x^i\), where Latin letters \(i,j,k,\dots \) from the middle of the alphabet will run over the range \(1,2,\dots ,d-1\). The summation convention will also be used for such spatial indices.

• • The inverse Minkowski metric \(\eta ^{\mu \nu }\) is defined by

  \(\seteqnumber{0}{1.}{14}\)

  \begin{flalign} \eta ^{\mu \nu }\,\eta _{\nu \rho } = \delta ^\mu _\rho = \eta _{\rho \nu } \,\eta ^{\nu \mu }\quad . \end{flalign} Its associated matrix \(\eta ^{-1}\) has the same form as \(\eta \).

• • A Poincaré transformation is a coordinate transformation of the form

  \(\seteqnumber{0}{1.}{15}\)

  \begin{flalign} x^{\prime \mu } = \Lambda ^\mu _{~~\nu }\,x^\nu + b^\mu \quad , \end{flalign} where \(b^\mu \) is a constant \(d\)-vector and \(\Lambda ^{\mu }_{~~\nu }\) is a constant \(d\times d\)-matrix that satisfies the condition

  \(\seteqnumber{0}{1.}{16}\)

  \begin{flalign} \label {eqn:metricPoincare} \eta _{\mu \nu }\,\Lambda ^{\mu }_{~~\rho }\,\Lambda ^{\nu }_{~~\sigma } = \eta _{\rho \sigma }\quad . \end{flalign} For \(b^\mu =0\), one calls \(x^{\prime \mu } = \Lambda ^\mu _{~~\nu }\,x^\nu \) a Lorentz transformation. Unless stated otherwise, we shall always assume that \(\det (\Lambda )=+1\) and that the entry \(\Lambda ^0_{~~0}>0\) is positive. This special class of Lorentz/Poincaré transformations is called proper and orthochronous. It excludes for example the time-reversal transformation \((t,\mathbf {x})\mapsto (-t,\mathbf {x})\), which is studied separately in QFT.

• • Defining

  \(\seteqnumber{0}{1.}{17}\)

  \begin{flalign} \Lambda _{\mu }^{~~\nu } := \eta _{\mu \rho }\,\eta ^{\nu \sigma }\,\Lambda ^{\rho }_{~~\sigma } \end{flalign} in terms of raising and lowering indices, one obtains a new matrix that satisfies

  \(\seteqnumber{0}{1.}{18}\)

  \begin{flalign} \label {eqn:inversemetricPoincare} \eta ^{\mu \nu }\,\Lambda _{\mu }^{~~\rho }\,\Lambda _{\nu }^{~~\sigma } = \eta ^{\rho \sigma }\quad . \end{flalign}

• • Tangent vectors \(v^\mu \) and cotangent vectors \(h_\mu \) on the Minkowski spacetime transform under Poincaré transformations as

  \(\seteqnumber{0}{1.}{19}\)

  \begin{flalign} v^{\prime \mu } = \Lambda ^{\mu }_{~~\nu }\,v^\nu \quad ,\qquad h_\mu ^\prime = \Lambda _{\mu }^{~~\nu }\,h_\nu \quad . \end{flalign} Note that, in contrast to the coordinates \(x^\mu \), there is no translation term \(b^\mu \) in these transformation formulas. The following contractions

  \(\seteqnumber{0}{1.}{20}\)

  \begin{flalign} h\, v := h_\mu \,v^\mu \quad ,\qquad v\, w := \eta _{\mu \nu } \,v^\mu \,w^{\nu } \quad ,\qquad h\, l := \eta ^{\mu \nu }\,h_\mu \,l_\nu \quad \end{flalign} among tangent vectors \(v^\mu ,w^\mu \) and cotangent vectors \(h_\mu ,l_\mu \) are Poincaré invariant. Splitting tangent and cotangent vectors into their time and space parts, these contractions read as

  \(\seteqnumber{1}{1.22}{0}\)

  \begin{flalign} h\, v &= h_0\,v^0 + \mathbf {h}\,\mathbf {v} = h_0\,v^0 + h_i\,v^i\quad ,\qquad \\ v\, w &= -v^0\,w^0 + \mathbf {v}\,\mathbf {w} = -v^0\,w^0 + \delta _{ij}\,v^i\,w^j\quad ,\qquad \\ h\, l &= -h_0\,l_0 + \mathbf {h}\,\mathbf {l} =-h_0\,l_0 + \delta ^{ij}\,h_i\,l_j\quad , \end{flalign} where the minus signs arise from \(\eta _{\mu \nu }\) and \(\eta ^{\mu \nu }\).

Fourier transform:

• • We define the Fourier transform of a function \(f\) on an \(n\)-dimensional Cartesian space \(\bbR ^n\) with coordinates \(\mathbf {x} = (x^1,\dots , x^n)\) by

  \(\seteqnumber{0}{1.}{22}\)

  \begin{flalign} \label {eqn:Fourier} \widetilde {f}(\mathbf {k}) := \int _{\bbR ^n} f(\mathbf {x})\, e^{-\ii \, \mathbf {k}\,\mathbf {x}}\,\dd \mathbf {x}\quad , \end{flalign} where \(\mathbf {k} = (k_1,\dots , k_n)\) denote the Fourier momenta, \(\mathbf {k}\,\mathbf {x} := k_i\, x^i\) is obtained by summation over \(i=1,\dots ,n\), and \(\dd \mathbf {x} := \dd x^1\cdots \dd x^n\) is the \(n\)-dimensional volume element.

• • The Fourier transform can be inverted via the inverse Fourier transform

  \(\seteqnumber{0}{1.}{23}\)

  \begin{flalign} \label {eqn:inverseFourier} f(\mathbf {x}) = \int _{\bbR ^n} \widetilde {f}(\mathbf {k})\, e^{\ii \,\mathbf {k}\,\mathbf {x}}\,\frac {\dd \mathbf {k}}{(2\pi )^n}\quad . \end{flalign} In fact, inserting (1.23) into (1.24), we compute

  \(\seteqnumber{0}{1.}{24}\)

  \begin{flalign} \nn \int _{\bbR ^n} \widetilde {f}(\mathbf {k})\, e^{\ii \,\mathbf {k}\,\mathbf {x}}\,\frac {\dd \mathbf {k}}{(2\pi )^n} &= \int _{\bbR ^n} \int _{\bbR ^n} f(\mathbf {y})\, e^{\ii \,\mathbf {k}\,(\mathbf {x}-\mathbf {y})}\, \dd \mathbf {y}\,\frac {\dd \mathbf {k}}{(2\pi )^n}\\ \nn &=\int _{\bbR ^n} f(\mathbf {y})\,\bigg (\int _{\bbR ^n} e^{-\ii \,\mathbf {k}\,(\mathbf {y}-\mathbf {x})} \,\frac {\dd \mathbf {k}}{(2\pi )^n} \bigg )\,\dd \mathbf {y}\\ &=\int _{\bbR ^n} f(\mathbf {y})\,\delta (\mathbf {y}-\mathbf {x})\,\dd \mathbf {y} = f(\mathbf {x})\quad , \end{flalign} where in the third step we have used that

  \(\seteqnumber{0}{1.}{25}\)

  \begin{flalign} \delta (\mathbf {y}-\mathbf {x}) = \int _{\bbR ^n} e^{-\ii \,\mathbf {k}\,(\mathbf {y}-\mathbf {x})} \,\frac {\dd \mathbf {k}}{(2\pi )^n} \end{flalign} is one of the many presentations of the \(n\)-dimensional Dirac delta function.