Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 43 / 3

Measure the statistical Hamiltonian in units of $k_{B}T$ so that the weight is $e^{-H_0}$, and take $\alpha >0$, $m^2>0$ for convergence. Use the Fourier transform conventionThe last identity is the reality condition, so modes at $p$ and $-p$ are not independent complex fields. Orthogonality of the Fourier exponentials gives $\int d^{D}x{\varphi }^2={\int }_p|\tilde{\varphi }(p){|}^2$, while a derivative multiplies a Fourier mode by $ip$. HenceThe kernel is positive for the stated parameters. Keeping an explicit inverse temperature instead would replace this kernel in the statistical weight by ${\beta }_{\mathrm{th}}\tilde{\Delta }$; the following source convention absorbs that factor.

Couple a real source through $H_J=H_0-\int d^{D}xJ(x)\varphi (x)$. Its Fourier components obey $\tilde{J}(-p)=\tilde{J}(p{)}^{\ast }$, and the source term is $-{\int }_p\tilde{J}(p)\tilde{\varphi }(p{)}^{\ast }$. Complete the square using the positive quadratic kernel from the previous part:Translation of the integration variables in the regulated Gaussian functional integral therefore givesThis also fixes the sign: a real linear source increases the Gaussian generating function by a positive quadratic exponent, unlike an imaginary Fourier-characteristic-function source.

To specify $Z_G$ rather than conceal an infinite normalization, first impose finite volume and an ultraviolet cutoff, and choose orthonormal real coordinates for the independent field modes. If their number is $N_R$ and the eigenvalues of the quadratic form are ${\Delta }_j>0$, the measure ${\prod }_{j=1}^{N_R}d{\varphi }_j$ givesA measure ${\prod }_{j}d{\varphi }_j/\sqrt{2\pi }$ removes the prefactor. In the continuum one may express this as $\log Z_G=-\frac{1}{2}\mathrm{Tr}\log \Delta$ plus a source-independent measure constant, with the same regulator understood. The real-mode prescription prevents double counting the conjugate Fourier pairs. All normalized correlation functions follow from $Z_0[J]/Z_G$ and do not depend on that constant.

The connected generating functional is $W_0[J]=\log Z_0[J]$. Its second functional source derivative gives the connected correlation function; the source-independent determinant drops out. Translational invariance therefore givesThis covariance is independent of the source in a purely Gaussian theory; at zero source the mean field vanishes.

Fourier inversion can be evaluated with the Schwinger parameterization. Since $(p^2+{\mu }^2{)}^{-1}={\int }_0^{\infty }ds{e}^{-s(p^2+{\mu }^2)}$, the elementary Fourier transform of a Gaussian giveswhere $K$ is the Modified Bessel function of the second kind. This integral formula also gives both asymptotic regimes directly.

For $\mu r\ll 1$, set $u=r^2/(4s)$. The integral tends to the gamma function integral ${\int }_0^{\infty }{u}^{D/2-2}{e}^{-u}du=\Gamma (D/2-1)$; this is finite for $D>2$. Hence, at distances larger than the microscopic cutoff,For $\mu r\gg 1$, the exponent ${\mu }^{2}s+r^2/(4s)$ has its minimum at $s=r/(2\mu )$, with value $\mu r$. Gaussian expansion about that saddle, or the large-argument asymptotic expansion of a modified Bessel function, gives $K_{\rho }(z)\sim \sqrt{\pi /(2z)}{e}^{-z}$. ThereforeThis is the massive Gaussian field correlation tail with its normalization displayed. For example $D=3$ gives exactly $G_0(r)=\alpha e^{-r/\xi }/(4\pi r)$, verifying both regimes and the exponential length scale.

Split the Fourier modes into $|p|<\Lambda /b$ and the shell $\Lambda /b<|p|<\Lambda$. The source is uniform, so it couples only to the zero mode, which belongs to the retained part. The Gaussian field theory has no couplings between the shell and retained modes. Integrating out the shell therefore contributes only a source-independent determinant to the free energy, without changing the quadratic coefficients of the retained Hamiltonian.

Restore the cutoff by $x=b{x}^{\prime }$ and choose the field normalization ${\varphi }_{<}(b{x}^{\prime })=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$. Counting the measure, derivative and field factors explicitly givesThus in the convention that preserves the kinetic normalization, the Gaussian momentum-shell scaling isThe field has engineering dimension $(D-2)/2$ and zero anomalous dimension. The measure Jacobian and shell determinant supply the additive constant described in Q2. A different choice of field rescaling can redistribute factors into $\alpha$, but must give the same physical transformed correlation length ${\xi }^{\prime }=\xi /b$.

At $m^2=0$ and $h=0$ the model is at a Gaussian fixed point. Its connected correlation function has the power-law critical form $r^{-(D-2)}$ rather than a finite exponential decay length. Approaching from $m^2>0$, the uniform susceptibility ${\tilde{G}}_0(0)=1/m^2$ and the correlation length both diverge. If the thermal scaling field is linear in $m^2$ with $\alpha$ finite and positive, thenThe same result follows from the mass multiplier $b^2$: its logarithmic thermal eigenvalue is two, so $\nu =1/2$.

The determinant also shows a thermodynamic singularity. Two derivatives of the free-energy density with respect to $m^2$ give, up to sign and a nonzero normalization,as $m^2\downarrow 0$. Rescaling $p=\sqrt{\alpha m^2}q$ establishes the power, and the dimensionless integral is ultraviolet-convergent for $D<4$. Integrating back gives a nonanalytic free-energy term proportional to $(m^2{)}^{D/2}$ after regular backgrounds are removed. This is critical behavior of the quadratic model.

There is a stability qualification: a pure Gaussian potential with $m^2<0$ is unbounded below in the uniform mode, so it does not by itself provide a stable ordered phase on the other side. The ordered-phase obstruction in a Gaussian scalar model is removed by adding a positive quartic interaction. In $2<D<4$ that interaction is relevant, so the resulting generic scalar critical point need not retain the Gaussian exponents. Thus the calculation proves the Gaussian critical singularity and $\nu =1/2$, without claiming that the unmodified quadratic integral defines both phases of a physical ordered transition.