Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2024 / 1

For a Dirac field, the path integral is a Berezin integral over independent Grassmann fields $\psi$ and $\bar{\psi }$:The vacuum boundary conditions and the Feynman i-epsilon prescription are understood; $\mathcal{N}$ fixes the vacuum normalization. This is the vacuum generating functional without sources, rather than a finite-temperature partition function.

The Dirac adjoint transforms as $\bar{\psi }\mapsto \bar{\psi }{e}^{-i\alpha }$. For constant $\alpha$, ${\partial }_{\mu }(e^{i\alpha }\psi )=e^{i\alpha }{\partial }_{\mu }\psi$. Both the kinetic term and the Dirac mass term therefore retain their values. This proves the global phase symmetry, whose group is the unitary group $U(1)$.

For the local variation $\delta \psi =i\alpha (x)\psi$, $\delta \bar{\psi }=-i\alpha (x)\bar{\psi }$, the terms proportional to $\alpha$ cancel and the differentiated phase givesThus, for a compactly supported $\alpha$, integration by parts givesThis Dirac current is the Noether current in the convention $\delta S=-\int j^{\mu }{\partial }_{\mu }\alpha$. Using the convention with a plus sign reverses the Noether current and the conserved charge together. There is one independent global symmetry in field theory, with four spacetime components of its Noether current. The associated conserved charge is $Q=\int d^{3}x{j}^0$, with vacuum subtraction when required.

An operator equation in a path integral $\mathcal{O}(x)=0$ means that insertion of $\mathcal{O}(x)$ into every time-ordered correlation function vanishes when $x$ is separated from the other insertion points. At coincident points, Ward identity contact terms can remain; an operator equation does not assert that these distributions vanish.

Let $X$ be a product of local insertions. Make the local phase change of integration variables from part b. The functional measure has unit functional Jacobian: the opposite phases of $\psi$ and $\bar{\psi }$ cancel in a vector-symmetry-preserving regularization in quantum field theory. There is no vector gauge anomaly. Invariance under a change of integration variables givesIf $X={\prod }_r{\mathcal{O}}_r(x_r)$ and ${\delta }_{\alpha }{\mathcal{O}}_r=i{q}_r\alpha (x_r){\mathcal{O}}_r$, comparison of the coefficients of the arbitrary $\alpha (x)$ yields the Ward-Takahashi identityConsequently ${\partial }_{\mu }{j}^{\mu }=0$ is an operator equation at separated points. This is stronger than just $⟨{\partial }_{\mu }{j}^{\mu }⟩=0$. For example, $X=\psi (y)\bar{\psi }(z)$ has $q_y=1$, $q_z=-1$, so its two Ward identity contact terms are $[-{\delta }^{(4)}(x-y)+{\delta }^{(4)}(x-z)]⟨T\psi (y)\bar{\psi }(z)⟩$; dropping them would give a false identity. Equivalently, the Dirac equation and the adjoint Dirac equation give ${\partial }_{\mu }(\bar{\psi }{\gamma }^{\mu }\psi )=0$, with the same coincident-point qualification in time-ordered correlation functions.

Couple a nondynamical background gauge field $B_{\mu }$ to the Dirac current, choosing a plus sign for the source term:There is no integration over $B_{\mu }$. The gauge covariant derivative is ${\partial }_{\mu }-i{B}_{\mu }$, so the simultaneous transformations areThe kinetic term changes by $-j^{\mu }{\partial }_{\mu }\alpha$ and the source term changes by $+j^{\mu }{\partial }_{\mu }\alpha$. The action is therefore invariant, exactly for finite $\alpha$ as well as infinitesimally. Changing integration variables and using the invariant functional measure from part c proves the background gauge invariance of a current generating functional:The allowed transformations preserve the vacuum boundary conditions. Its infinitesimal version is ${\partial }_{\mu }\delta Z/\delta B_{\mu }=0$. Because $B_{\mu }$ is only a source, this statement needs no gauge fixing or photon kinetic term. The absence of a vector gauge anomaly is essential to the change-of-variables argument.

Collect the two Dirac fields into $\Psi =({\psi }_1,{\psi }_2{)}^T$, with a common nonzero mass $m$. ThenAmong complex-linear rotations of the Dirac fields, the internal symmetry of a classical field theory is $\Psi \mapsto U\Psi$, $U\in U(2)$, with $\bar{\Psi }\mapsto \bar{\Psi }{U}^{†}$. Unitarity preserves the kinetic term, and the mass matrix commutes with every $U$. This unitary flavour symmetry of degenerate Dirac fields has group $(SU(2)\times U(1))/{\mathbb{Z}}_2$, where the quotient avoids counting the common central element twice.

Choose the four Hermitian matrices $T^0=1_2/2$, $T^a={\sigma }^a/2$ for $a=1,2,3$, with ${\sigma }^a$ the Pauli matrices. Promoting the four parameters to functions gives $\delta S=-\int d^{4}x({\partial }_{\mu }{\alpha }^A)\bar{\Psi }{\gamma }^{\mu }{T}^A\Psi$. The four Noether currents are thereforeTheir divergences vanish by the equal-mass Dirac equations. The singlet is half the total Dirac current; the other three currents rotate the two species.

Free-theory qualification. If “global internal symmetry” allows mixing the Dirac fields with charge-conjugate fields, the completely free theory written here has a larger enhanced internal symmetry of free Dirac fields. Write ${\psi }_i=({\lambda }_{2i-1}+i{\lambda }_{2i})/\sqrt{2}$ using four Majorana fermions. Up to a total derivative,The full orthogonal group $O(4)$ acts on the four real species; its connected part is $SO(4)$. Its six Noether currents can be chosen as $J_{AB}^{\mu }=i{\bar{\lambda }}_A{\gamma }^{\mu }{\lambda }_B$ for $A<B$, up to the common sign convention. The antisymmetry of this Majorana fermion vector bilinear makes these the six independent rotations. A concrete rotation outside the stated complex-linear $U(2)$ is ${\lambda }_2\mapsto \cos \theta {\lambda }_2+\sin \theta {\lambda }_3$, ${\lambda }_3\mapsto -\sin \theta {\lambda }_2+\cos \theta {\lambda }_3$: it preserves the displayed Lagrangian density while mixing ${\psi }_i$ with their charge-conjugate fields. The usual $U(2)$ answer specifies the subgroup preserving the chosen common fermion number.

In the Weyl representation of the gamma matrices, write each Dirac field as ${\psi }_i=({\ell }_{i\alpha },r_i^{\dot{\alpha }}{)}^T$, where ${\ell }_i$ and $r_i$ have opposite chirality. With the Weyl sigma matrices ${\sigma }^{\mu }=(1,\mathbf{\sigma })$ and ${\bar{\sigma }}^{\mu }=(1,-\mathbf{\sigma })$,The Dirac mass term couples the two Weyl fields. Under $\ell \mapsto U_L\ell$, $r\mapsto U_{R}r$, its first term acquires $U_L^{†}{U}_R$, so a common nonzero mass requires $U_L=U_R$. At zero mass there is no such constraint:This is the chiral flavour symmetry of massless Dirac fields. Its eight Noether currents are $j_{L,A}^{\mu }={\ell }^{†}{\bar{\sigma }}^{\mu }{T}^A\ell$ and $j_{R,A}^{\mu }=r^{†}{\sigma }^{\mu }{T}^{A}r$. The sum $j_R+j_L$ is the vector current, and $j_R-j_L$ is the axial current, for each flavour generator. This statement concerns the free theory in flat spacetime; an interacting theory can have a chiral anomaly.

For completeness, allowing the extra transformations described in part e gives four left-handed Weyl fields, $q=({\ell }_1,{\ell }_2,r_1^c,r_2^c{)}^T$, where $r_i^c$ is the left-handed charge-conjugate field of $r_i$. After integration by parts their kinetic term is $i{q}^{†}{\bar{\sigma }}^{\mu }{\partial }_{\mu }q$, whose maximal linear unitary group is $U(4)$. The symmetric mass matrix for equal Dirac masses iswith the antisymmetric spinor contraction implicit. A unitary change of basis makes $M=m{1}_4$; preserving both the kinetic term and $M$ requires $V^{†}V=V^{T}V=1_4$, hence real $V\in O(4)$. Thus the unrestricted free-theory enhancement is $U(4)$ at zero mass and $O(4)$ at equal nonzero mass; the familiar $U(2{)}_L\times U(2{)}_R$ and diagonal $U(2)$ are their common-fermion number-preserving subgroups.