Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 62 (Advanced Quantum Field Theory)

Use canonical quantization with $[\hat{q},\hat{p}]=i\hslash$, $\hat{p}=-i\hslash {\partial }_q$, and $\hat{H}={\hat{p}}^2/(2m)+V(\hat{q})$. Assume a real potential energy with the regularity and lower-bound conditions needed for a self-adjoint Hamiltonian and its Trotter product formula; singular potentials require their domain and limiting prescription to be specified separately. The transition amplitude is $K(\beta ,\alpha ;T)=⟨\beta |e^{-iT\hat{H}/\hslash }|\alpha ⟩$.

Take $\delta =T/N$, $q_0=\alpha$, $q_N=\beta$, and choose the left-endpoint potential energy convention. The precise time-sliced phase-space path integral isThe endpoint positions are fixed and the momenta are unconstrained. Its continuum notation iswhere the functional measure means the displayed limit, not an unspecified product of unnormalized measures.

At each slice, complete the square in momentum. With the square-root branch selected by continuation from positive Euclidean time,Thus the time-sliced equivalence of phase-space and configuration-space path integrals holds already at finite $N$:This defines the configuration-space path integralThere are $N$ Gaussian prefactors but only $N-1$ coordinate integrations. Omitting that distinction would change the kernel normalization. The paths integrated in the limit need not be differentiable; the kinetic action notation stands for the lattice difference expression.

To identify this with the operator amplitude, normalize $⟨q|p⟩=(2\pi \hslash {)}^{-1/2}{e}^{ipq/\hslash }$. The kernel of one product factor is exactlyInsert $N-1$ position resolutions of the identity between factors. Their product is precisely $K_N$. The Trotter product formula then givesso both functional integrals represent the same canonical kernel. One can define the finite integrals first at $T=-i\tau$, $\tau >0$, where the Gaussian momentum and coordinate kernels converge, and then continue to the real-time boundary value. Alternatively use an equivalent oscillatory damping prescription. Operator convergence fixes the distributional kernel limit; it need not imply pointwise convergence at every endpoint for every admissible potential energy. There is no operator-ordering ambiguity for this separated kinetic-plus-potential Hamiltonian once the common slicing convention is fixed.

Use Minkowski spacetime with signature $(+,-,\dots ,-)$ and $\hslash =1$, with a vacuum time-ordering prescription. At a regulator, define the normalized generating functionalThe normalized vacuum correlation functions of a time-ordered product areThe connected generating functional isThe logarithm selects connected source diagrams, or equivalently cumulants. With the assumed vanishing one-point function, derivatives of $Z=e^{iW}$ give the centered four-point cumulant decomposition:In detail, four differentiations either act on one connected block or form one of three pair partitions. Contributions with blocks of size one vanish. If $W_{ij}$ and $W_{1234}$ denote source derivatives at zero, the same relation is $G_{1234}=i{W}_{1234}-W_{12}{W}_{34}-W_{13}{W}_{24}-W_{14}{W}_{23}$, since $C_2=-i{W}_2$ and $C_4=i{W}_4$.

Let $\phi (x)=\delta W/\delta J(x)$ be the source-dependent mean field. On a locally invertible source-to-field branch, define the quantum effective action by the Minkowski Legendre transformIts variation is $\delta \Gamma =-\int J\delta \phi$, so $\delta \Gamma /\delta \phi =-J$. Differentiating the two inverse source-field maps yields the Minkowski inverse-Hessian relation for an effective action:Thus ${\Gamma }^{(2)}=-(W^{(2)}{)}^{-1}$. The inverse is understood at the regulator and on a nonsingular fluctuation sector; it is not an assertion that every source-field map is globally invertible.

For a free real scalar field, integrate by parts to writewhere repeated spacetime variables are integrated. Specify the propagator convention explicitly:Completing the Gaussian path integral around $\varphi =-A_{ϵ}^{-1}J$ cancels the source-independent determinant between numerator and denominator. ThereforeThe positive imaginary term in $A_{ϵ}$ damps the oscillatory Gaussian. Together with vacuum projection at the time boundaries, it selects the Feynman poles and the time-ordered, in-out correlation function rather than a retarded inverse. It can also be obtained by continuation from the Euclidean vacuum integral. A denominator convention without the numerator $i$ redistributes the displayed factors of $i$.

Finally $\phi =-A_{ϵ}^{-1}J$, hence $J=-A_{ϵ}\phi$. Substitution into the Legendre transform gives the vacuum-normalized effective action of a free scalar field:All connected functions beyond order two vanish and all proper vertices beyond the classical quadratic kernel vanish. Vacuum normalization fixes the additive constant; without it, the quantum effective action equals the classical action up to a field-independent determinant contribution.

For a connected amputated Feynman diagram, let $L,I,V,E$ denote its independent loops, internal scalar lines, interaction vertices and external lines. Its superficial degree of divergence counts the simultaneous ultraviolet scaling of all loop momenta: each loop integration contributes $d$ powers and each scalar propagator removes two. A nonderivative interaction adds no numerator powers, so $\Delta =dL-2I$. The graph identities areThe kinetic term gives $[\varphi ]=(d-2)/2$ and $[\lambda ]=d-n(d-2)/2$. Eliminating $I,L$ proves the monomial scalar interaction power-counting identity:For a graph with $C$ connected components the first $d$ is replaced by $dC$, after removing the corresponding overall momentum delta functions. Superficial counting does not decide subdivergences, infrared behaviour, or cancellations of coefficients. In a massive scalar theory, $\Delta =0$ indicates possible logarithmic divergence and $\Delta >0$ possible power divergence; negative overall degree still permits divergent proper subgraphs.

Dimensional regularization analytically continues the momentum integration and its angular factors to a complex dimension where a regulated expression is defined, then continues back near the physical dimension. Introduce a scale $\mu$ to keep the renormalized coupling in fixed units. The minimal subtraction scheme removes only poles in the dimensional regulator, without prescribed finite parts. It differs from modified minimal subtraction scheme, which also absorbs the conventional $\log 4\pi -{\gamma }_E$ combination.

For the cubic calculation, write $g=3!\lambda$, so the interaction is $-g{\varphi }^3/3!$ and the vertex is $-ig$. Use $D=6-2ϵ$ and the vertex $-ig{\mu }^{ϵ}$. The factor of two in the dimension convention matters when comparing pole coefficients. In six dimensions $[\varphi ]=2$, $[g]=0$ and a proper two-point graph has $\Delta =2$.

The one-loop bubble diagram has Feynman-diagram symmetry factor two. Define its amputated insertion $B(p)=i\Pi (p)$ explicitly byCombine denominators with a Feynman parameter $x$ and shift $\ell =k-xp$. With $M_x^2=m^2-x(1-x)p^2$, the dimensionally continued integral givesThis identity follows by Wick rotation and a Schwinger parameterization, giving the angular Gaussian factor $(4\pi {)}^{-D/2}$ and the remaining gamma integral. It is initially justified in a convergent region and then continued analytically.

Here $\Gamma (ϵ-1)=-1/ϵ+O(1)$ and ${\int }_0^{1}x(1-x)dx=1/6$. Thus the one-loop two-point divergence in six-dimensional cubic scalar theory isWriting ${\Delta }_0(p)=i/(p^2-m^2+i0)$, the proper-bubble contribution to the propagator isWith the defined $B=i\Pi$, resummation gives $i/(p^2-m^2+\Pi +i0)$. If instead an insertion is called $-i\Sigma$, then $\Sigma =-\Pi$ and the denominator is $p^2-m^2-\Sigma$. These naming conventions must not reverse the calculated graph.

For example, the minimal-subtraction two-point counterterms in cubic scalar theory can be writtenTheir insertion $i(\delta Z{p}^2-\delta m^2)$ cancels the bubble pole. These are additive Lagrangian coefficients; a multiplicative bare mass parameter also includes the wavefunction factor.

There is a vacuum convention to specify because a cubic interaction does not preserve a zero mean field automatically. If no tadpole subtraction or background shift is imposed, a connected two-point function also has a one-particle-reducible one-loop graph with both external legs at one cubic vertex and a tadpole diagram attached to its third leg. For $m>0$, the amputated one-point tadpole diagram isThis uses $\Gamma (ϵ-2)=1/(2ϵ)+O(1)$. Its mean-field pole is $v_{\mathrm{div}}={\Delta }_0(0)(T_1{)}_{\mathrm{div}}=-g{m}^2/[4(4\pi {)}^3ϵ]$. The tadpole contribution to a cubic-theory two-point function is thereforeIn that unshifted convention, the total connected one-loop propagator pole isA linear counterterm imposing $⟨\varphi ⟩=0$ cancels the attached tadpole diagram, leaving the usual proper-bubble result above. Vacuum bubbles cancel by normalization. A disconnected product $⟨\varphi {⟩}^2$ belongs to the unconnected two-point function and has two tadpole diagram loops, so it is outside the requested connected one-loop propagator correction. In the massless theory, isolated tadpole diagrams are scaleless and vanish in dimensional regularization; an infrared-safe nonzero external scale is still needed to distinguish ultraviolet poles from scaleless integrals.

Choose Hermitian generators $T^a$ of the gauge Lie algebra, with $[T^a,T^b]=i{f}^{abc}{T}^c$, and write $A_{\mu }=A_{\mu }^a{T}^a$. For the usual compact gauge algebra choose $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$. More generally use a nondegenerate invariant symmetric bilinear form on the algebra in place of this trace contraction; a completely arbitrary component metric would not give gauge invariance.

A matter vector is useful to fix conventions: with ${\psi }^{\prime }=U(x)\psi$ and $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$, require $D_{\mu }^{\prime }{\psi }^{\prime }=U{D}_{\mu }\psi$. Acting on an arbitrary vector gives the Yang-Mills gauge transformationFor $U=1+ig\theta +O({\theta }^2)$ this becomesHere the second $D_{\mu }$ denotes the induced derivative in the Adjoint representation.

Define the curvature by $[D_{\mu },D_{\nu }]=-ig{F}_{\mu \nu }$. Expanding the commutator derivesSince commutators of the transformed derivatives conjugate, $F_{\mu \nu }^{\prime }=U{F}_{\mu \nu }{U}^{-1}$. Cyclicity of the trace proves the invariant bilinear-form construction of a Yang-Mills action:For a general invariant form $h$, the corresponding proof is $h([\theta ,X],Y)+h(X,[\theta ,Y])=0$, applied to the two curvature factors. No matter field is required in this pure-gauge Lagrangian.

Introduce the adjoint Faddeev-Popov ghost $c^a$ and antighost ${\bar{c}}^a$, independent Grassmann-odd scalar fields of ghost numbers $+1$ and $-1$, together with the Grassmann-even Nakanishi-Lautrup field $b^a$ of ghost number zero. The bar labels the independent antighost, not an additional propagating complex-conjugate matter field. Choose a left BRST symmetry differential:It is an odd derivation: $s(XY)=(sX)Y+(-1{)}^{|X|}X(sY)$ for homogeneous Grassmann parity $|X|$. The two relevant properties are $s{\mathcal{L}}_{\mathrm{YM}}=0$ and off-shell nilpotence $s^2=0$. The first is gauge invariance with parameter replaced by the ghost, and the second follows from the Lie algebra's graded Jacobi identities. ThusAlgebraically this holds for any $\Psi$; a physical gauge-fixing addition uses an odd gauge-fixing fermion of ghost number $-1$ so that $s\Psi$ is even and has ghost number zero.

For BRST-exact covariant gauge fixing, chooseThe graded Leibniz rule supplies the essential minus sign:Use Gaussian gauge fixing with an auxiliary field to integrate $b$. Pointwise,Translation of the regulated oscillatory Gaussian leaves only a field-independent normalization. The remaining Lagrangian is thereforeThis is covariant Feynman gauge, with gauge parameter one. Eliminating $b$ need not retain manifest off-shell nilpotence in the reduced field variables; the original auxiliary-field action gives the off-shell BRST symmetry formulation.

Finally, the ungauge-fixed quadratic momentum kernel isGauge directions are null vectors, so there is no inverse defining a vector propagator, and the functional integral also counts an infinite gauge orbit volume. The added gauge-fixing term cancels the longitudinal part, leaving $Q_{\mu \nu }^{ab}=-{\delta }^{ab}{p}^2{\eta }_{\mu \nu }$. With the vacuum prescription, the propagators areThe ghost action represents the Faddeev-Popov determinant and supplies ghost-vector vertices; closed ghost loops have the fermionic minus sign. Expanding the cubic and quartic Yang-Mills terms and this ghost interaction now defines consistent perturbative Feynman rules. Physical mass-shell poles remain, whereas the gauge degeneracy has been removed. Possible global gauge-fixing obstructions and boundary zero modes are separate from the perturbative inverse around the trivial vacuum.