 Solution

The renormalization group organizes how a statistical system changes when observed at progressively larger length scales. Start with its partition function and a microscopic cutoff $\Lambda$. Split the Fourier transform field into modes below $\Lambda /b$ and modes in the shell $\Lambda /b<|q|<\Lambda$. Integrate the shell modes, rescale coordinates $x^{\prime }=x/b$ to restore the cutoff, and rescale the field to keep a chosen gradient normalization. This is a renormalization-group transformation. It preserves the partition function when the generated interactions and additive free energy terms are retained; truncating the resulting operator expansion is an approximation.

A renormalization-group fixed point has an invariant dimensionless interaction structure. The continuous flow of a nearby scaling field $v_i$ takes the linearized formA positive $y_i$ defines a relevant operator, a negative $y_i$ an irrelevant operator, and a zero $y_i$ a marginal operator requiring nonlinear analysis. The continuous-flow eigenvalues $y_i$ should not be confused with the discrete multipliers $b^{y_i}$. To reach a critical point, tune the relevant thermal variable and set the relevant symmetry-breaking source to zero. The resulting flow stays on the critical surface.

Microscopically different systems whose remaining couplings approach the same fixed point share a universality class. Irrelevant perturbations lose their influence on long-distance exponents. Relevant directions explain why varying the temperature or field moves the system away from criticality. The thermal scaling field obeys $t^{\prime }=b^{y_t}t$, while the rescaled correlation length is ${\xi }^{\prime }=\xi /b$. Taking $b=|t{|}^{-1/y_t}$ therefore gives $\nu =1/y_t$.

If the field has scaling dimension $x_{\varphi }=(d-2+\eta )/2$, the source term $\int h\varphi$ has eigenvalue $y_h=d-x_{\varphi }=(d+2-\eta )/2$. The singular free-energy density then obeyswhen the hyperscaling relation is valid. This derives the scaling hypothesis for critical phenomena and the exponent relations of Question 2 from the fixed-point picture.

At the Gaussian fixed point, keeping $\int (\nabla \varphi {)}^2$ invariant gives $x_{\varphi }=(d-2)/2$. The quadratic coupling has $y_r=2$ and the quartic coupling has $y_u=4-d$. Thus quartic interactions are relevant below four dimensions, marginal at four, and irrelevant above four. Below four, in dimensions where an ordinary continuous short-range scalar transition exists, an interacting Wilson-Fisher fixed point governs that transition. Above four, the Gaussian fixed point gives mean-field powers, but the quartic term remains necessary to stabilize the ordered state: it is a dangerously irrelevant coupling, explaining the failure of naive hyperscaling. The renormalization group connects scale invariance, universality, relevant tuning parameters and fluctuation corrections in one framework.