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Write ${\beta }_T=(k_{B}T{)}^{-1}$ for inverse temperature to distinguish it from the order-parameter critical exponent. A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ is a nonnegative conditional distribution of coarse spins given the fine spins, normalized byFor deterministic blocking it can impose a majority rule or block average, with a Dirac delta replacing the sum constraint for continuous coarse variables. Each block occupies $b^D$ original cells, so the new lattice spacing and number of sites areThe physical volume $N{a}^D$ is unchanged. After $p$ blockings, $a_p=b^{p}a$ and $N_p=b^{-pD}N$.

Define the coarse statistical Hamiltonian by the exact Boltzmann-weight identityThe operator family must include all generated interactions for this renormalization-group transformation to be exact; restricting it to a few couplings is an approximation. The scalar $g(u)$ has energy units and is the contribution per original site that is independent of ${\sigma }^{\prime }$. Restoring the original constant term $NC$ giveshenceThis convention measures $g$ per original site; defining a constant per new site would move a factor $b^D$ into its definition.

The identity-operator contribution to renormalization-group free energy $C_p$ records the configuration-independent normalization generated by the eliminated spins. It includes their contribution to free energy, including entropy and measure factors, and need not be independent of temperature. It has no effect on normalized probabilities of the retained spins, but is indispensable for the total partition function and its temperature derivatives. Iterating its recursion givesThus $N_p{C}_p$ is the original constant $N{C}_0$ plus the accumulated free-energy contributions of the eliminated degrees of freedom.

To obtain the additive free-energy recursion under blocking, use energy-valued free energy per current lattice site,The blocking kernel normalization impliesSubstituting the recursion for $C^{\prime }$ cancels $C$, givingA second step gives $f_0(u_0)=b^{-2D}{f}_0(u_2)+g(u_0)+b^{-D}g(u_1)$, and induction provesFor dimensionless free energy $-\log Z/N$, multiply $F,f_0,g,C$ throughout by ${\beta }_T$.

The singular free-energy density must be distinguished from the exact per-site representative $f_0$. If $a(u)$ is a regular background and $f_s=f_0-a$, thenThus the same displayed inhomogeneous recursion holds with $f_s,g_s$ after subtraction. When a suitable analytic background solves $a(u)=g(u)+b^{-D}a(Ru)$, it leaves a homogeneous singular recursion. Analytic subtraction of an inhomogeneous renormalization recursion explains this separation. A renormalization-group free-energy resonance can obstruct analytic subtraction and produce a logarithm. The original function $g$ originates in the short-distance spins that were summed out, rather than in an extra external interaction.

Near a renormalization-group fixed point $u_{\ast }$, linearize the transformation:An eigenvector of $\mathsf{A}$ with eigenvalue $b^{y_i}$ defines a scaling coordinate $t_i$ with $t_i^{\prime }=b^{y_i}{t}_i$ to leading order. A relevant operator has $y_i>0$: its coupling grows towards large distances, so it must be tuned to reach the fixed point. An irrelevant operator has $y_i<0$ and dies away; a marginal operator has $y_i=0$ and requires higher-order flow analysis. An operator of scaling dimension $x_i$ has $y_i=D-x_i$.

The coarse spins also require a multiplicative normalization. In continuum notation, writewhere ${\varphi }_{<}$ is the retained field and $\eta$ is its anomalous dimension. This restores the normalization of the critical long-distance connected correlation function. Since a uniform field conjugate to an order parameter enters through $\int d^{D}xH\varphi$, it has eigenvalue exponent $y_h=D-x_{\varphi }=(D+2-\eta )/2$.

Let $t$ be the thermal relevant coordinate with exponent $y_t>0$. After regular backgrounds have been removed, the leading singular free-energy density obeysfor a large total scale factor $B=b^p$. Choosing $B=|t{|}^{-1/y_t}$ gives the scaling hypothesis for critical phenomenaTwo temperature derivatives give the heat-capacity critical exponent; one conjugate-field derivative gives the order-parameter critical exponent; two such derivatives give the magnetic-susceptibility critical exponent. Scaling the correlation length gives $\xi (t)=B\xi (B^{y_t}t)$. Consequently,At $t=0$, instead choose $B=|H{|}^{-1/y_h}$; differentiating gives $\delta =y_h/(D-y_h)$. These scaling relations for critical exponents assume that irrelevant variables can be set to their fixed-point values without singular consequences. A dangerously irrelevant coupling or logarithmic correction requires keeping the corresponding dependence.

For the Gaussian field theory, integrate out the Fourier modes in $\Lambda /b<|q|<\Lambda$ and retain ${\varphi }_{<}$. The modes are independent at zero external field, so this Gaussian thinning transformation changes the normalization by a Gaussian determinant and leaves a quadratic statistical Hamiltonian for the retained modes. Restore the cutoff withIn the integrated statistical Hamiltonian, the volume element contributes $b^D$, each gradient contributes $b^{-1}$, and each field contributes $b^{-(D-2)/2}$. Thus the coefficient ${\kappa }^{-1}$ of the gradient energy is unchanged, whileIn particular $y_t=2$, $y_h=(D+2)/2$ and $\eta =0$. The Gaussian critical exponents and the Gaussian order-parameter scaling index therefore give the requested indicesThe sign convention $+h\varphi$ corresponds to the physical conjugate field $H=-h$ and does not affect these powers.

There are physical qualifications to these Gaussian indices. The strictly quadratic Gaussian field theory is stable for positive $m^2$ and has no stable ordered phase for negative $m^2$; its purported spontaneous $\beta$ is therefore the formal Gaussian order-parameter scaling index. A stabilizing quartic term gives an actual ordered phase. Below four dimensions that term is relevant and changes the exponents; above four dimensions it is a dangerously irrelevant coupling and gives the actual mean-field $\beta =1/2$, rather than the Gaussian hyperscaling value.

The heat capacity power can also be checked directly. Apart from constants, the zero-field Gaussian free-energy density isIf $m^2\propto t$, its singular second temperature derivative is controlled by $\int d^{D}q(q^2+\kappa m^2{)}^{-2}$. For $0<D<4$, rescaling $q=\sqrt{\kappa m^2}k$ gives a power $t^{D/2-2}$, as above. At $D=4$ there is a logarithm, and for $D>4$ an analytic background dominates the heat capacity. The negative formal Gaussian $\alpha$ then describes the subleading singular piece, with logarithmic resonances at appropriate even dimensions. This is the Gaussian specific-heat infrared threshold, and it prevents confusing singular indices with the entire observable.

Take a positive gradient coefficient ${\kappa }^{-1}$, a positive squared mass, a finite-volume regulator and an ultraviolet cutoff. Use the dimensionless statistical action convention $Z=\int \mathcal{D}\varphi e^{-H}$: inverse temperature is absorbed into the coefficients. If $H$ denotes dimensional energy instead, the exponent below gains a factor ${\beta }_{\mathrm{th}}$ and the covariance gains ${\beta }_{\mathrm{th}}^{-1}$. These conventions cannot be mixed.

Choose the Fourier transform conventionIntegration of the quadratic gradient and source terms givesComplete the square with ${\tilde{\varphi }}_{\mathrm{cl}}(p)=\tilde{\Delta }(p{)}^{-1}\tilde{J}(p)$. Translation of the regulated real Gaussian variables leaves their measure unchanged, so the Gaussian integral givesHere $Z_G=Z_0[0]$ is the source-free Gaussian partition function. For $n$ independent regulated real coordinates and a positive quadratic matrix $A$, it is $(2\pi {)}^{n/2}(\det A{)}^{-1/2}$ under Lebesgue normalization. The continuum determinant needs the stated regulator and a measure convention. Reality pairs the nonzero Fourier modes, so they must not be counted as two independent real fields.

A connected correlation function subtracts the product of the means. The logarithm $W[J]=\log Z_0[J]$ generates its cumulants: differentiating $W$ twice with respect to the source, using the measure $d^{D}p/(2\pi {)}^D$ in the functional derivative convention, yieldsThe source changes the mean but not the covariance; the stated zero-source limit therefore follows immediately. Momentum conservation uses $p+q$, appropriate for a real field without complex conjugation on either displayed factor.

For the continuum massive Gaussian field correlation tail, set $\mu =\sqrt{\kappa }m>0$ and $r=|x|$. Fourier inversion givesThe large-argument Modified Bessel function of the second kind has $K_{\rho }(z)\sim \sqrt{\pi /(2z)}{e}^{-z}$. Hence $G_0(r)$ is proportional to $r^{-(D-1)/2}{e}^{-\mu r}$ at large distance andAt zero mass the exponential length diverges. For $D>2$ the critical continuum tail is algebraic, $G_0(r)\propto r^{2-D}$; in lower dimensions infrared regulation needs additional care.

For a uniform source $h$, perform a momentum-shell renormalization group step: separate slow modes $|p|<\Lambda /b$ and fast modes $\Lambda /b<|p|<\Lambda$. In a Gaussian field theory they decouple; the uniform source couples only to the retained zero mode. Integrating fast modes contributes only a field-independent determinant to the free energy. Restore the cutoff by $x^{\prime }=x/b$, $p^{\prime }=bp$ andThe factors from the measure, gradient and two fields cancel in the gradient term. The mass term gains $b^2$, and the linear source gains $b^{(D+2)/2}$. ThereforeHere $m$ is the positive mass; for signed thermal coordinates it is the squared-mass relation that applies. The Gaussian fixed point has thermal scaling eigenvalue ${\lambda }_t=2$, source eigenvalue ${\lambda }_h=(D+2)/2$ and zero anomalous dimension.

Under the formal homogeneous Gaussian scaling hypothesis, substitution of those eigenvalues givesThe last two values are independently visible in $\xi \propto t^{-1/2}$ and $\chi ={\tilde{G}}_0(0)=1/m^2\propto t^{-1}$, agreeing with Landau-Ginzburg theory. The specific-heat power follows directly from the determinant: after two thermal derivatives its singular contribution is proportional toThe change of variables $p=\sqrt{\kappa t}q$ proves this power because the rescaled integral converges at large $q$ exactly when $D<4$. The two-dimensional free energy itself has a $t\log t$ resonance, while its second derivative still has the indicated $t^{-1}$ power.

The printed magnetization exponent is a formal Gaussian scaling index, not a proved ordered-phase law of the purely quadratic model. The ordered-phase obstruction in a Gaussian scalar model is explicit: the uniform field has energy density $m^2{M}^2/2-hM$. At $h=0,m^2<0$ it tends to $-\infty$ as $|M|\to \infty$, and the partition function diverges. For $m^2>0$, the mean is $M=h/m^2$ and tends to zero as $h\to 0$, so there is no spontaneously ordered branch. Thus no literal $M\sim |t{|}^{\beta }$ for $t<0$ can be established from this Statistical Hamiltonian alone. The value $(D-2)/4$ follows from the scaling dimension of the field if one assumes such a branch; for $D\le 2$ its nonpositive value further warns against that interpretation. Adding a stabilizing quartic interaction defines an ordered phase, but below four dimensions that interaction is relevant and generally leads to the Wilson-Fisher fixed point, not these Gaussian thermodynamic powers.

For the Gaussian specific-heat infrared threshold, at $D=4$ the same determinant integral grows like $\log ({\Lambda }^2/t)$, so $\alpha =0$ as a power exponent, with a logarithmic divergence. For $D>4$, it has a finite cutoff-dependent limit; the leading heat capacity is a regular constant and has no divergent power, again described by $\alpha =0$ in the requested convention. After subtraction of this background the Gaussian singular term can still have the negative power index $2-D/2$; it is not correct to erase that distinction. In a stable quartic Landau-Ginzburg theory above four dimensions, the quartic coefficient is a dangerously irrelevant coupling: it is needed for the ordered phase and contributes $f_s\propto -t^2/u$, giving the mean-field specific-heat jump and $\alpha =0$ while invalidating naive hyperscaling relation. At four dimensions, marginal interactions can add further logarithmic corrections. The agreement with Landau theory is in the leading heat-capacity power; it does not assert the absence of logarithms or supply a missing stable Gaussian ordered phase.