 Solution

Renormalization compares the same long-distance physics at different observation scales. The starting object is a microscopic or cutoff statistical Hamiltonian ${\mathcal{H}}_g$ with every symmetry-allowed coupling $g$. Because a critical correlation length is much larger than a lattice spacing, repeatedly eliminating short-distance detail can reveal scale-invariant behavior.

In a real-space transformation choose a normalized blocking kernel $B(S|s)$ for retained variables. DefineSumming over $S$ proves exact preservation of $Z$ before truncation. The transformed Hamiltonian generally gains longer-range and many-body interactions and a field-independent constant. Observables must be blocked consistently. In momentum-shell language split $\varphi ={\varphi }_{<}+{\varphi }_{>}$, integrate $\Lambda /b<|q|<\Lambda$, then use $x^{\prime }=x/b$ and a field rescaling to restore cutoff and kinetic normalization. Coarse-graining, rescaling and tracking generated interactions are distinct steps.

Repeated transformations generate a flow $g^{\prime }=R_b(g)$ with composition $R_b{R}_c=R_{bc}$ when the same exact coarse-graining family is used. Losing degrees of freedom generally makes this a semigroup, despite its conventional group name. A renormalization-group fixed point is invariant after rescaling. At a continuous transition the large-distance theory approaches a critical renormalization-group fixed point; finite correlation length instead eventually flows into an ordinary thermodynamic phase regime.

Linearize in scaling coordinates $u_i$:For increasing observation length, $y_i>0$ is relevant and must be tuned, $y_i<0$ is irrelevant, and $y_i=0$ is marginal and needs nonlinear analysis. Discrete multipliers $b^{y_i}$ must be compared with one, not with zero. The stable manifold forms the critical surface after relevant directions have been set to zero.

After subtracting regular additive contributions, ordinary hyperscaling givesChoosing $b=|t{|}^{-1/y_t}$ or choosing the field as scale yieldsFor a critical two-point function $G(r)\sim r^{-(d-2+\eta )}$, the order parameter scaling dimension is $x_m=(d-2+\eta )/2$, and its conjugate field has $y_h=d-x_m=(d+2-\eta )/2$. This recovers the Q1 identities while distinguishing the gap exponent $\Delta$ from the field's scaling dimension $y_h$.

As a concrete dimensional check, start from a scalar gradient action. Restoring its coefficient uses ${\varphi }^{\prime }(x^{\prime })=b^{(d-2)/2}{\varphi }_{<}(b{x}^{\prime })$, so at the Gaussian fixed pointQuartic interactions are relevant below four dimensions. They can produce an interacting Wilson-Fisher fixed point, whose exponents differ from naive Gaussian counting. Above four, the quartic coupling is irrelevant at the Gaussian fixed point but dangerous for ordered observables: ${\varphi }_0^2=-r/u$ is singular when $u$ is dropped. Thus the simple two-variable hyperscaling formula is not universal above the upper critical dimension.

Different microscopic systems can flow to the same long-distance renormalization-group fixed point, explaining universality of exponents and normalized scaling forms. Critical temperatures, metric factors and regular free energies remain nonuniversal. Dimension, order parameter symmetry, interaction range and additional soft variables constrain the universality class. Truncations, perturbative expansions and numerics approximate the flow; they must not be confused with the exact elimination definition.^[ref]