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In a three-dimensional uniform isotropic continuum obeying the Principle of Galilean relativity, with one bare particle mass $m$, the Fermi-liquid effective mass satisfiesA band effective mass need not satisfy this relation.^[ref]

For the stipulated $m^{\ast }=2m$ in the occupation figure, this relation requires $F_1^s=3$ if those assumptions apply. At fixed $p_F$, $v_F$ is half its free-particle value, so moving by $0.04p_F$ from the Fermi surface costs approximately $0.02p_F^2/m$ instead of $0.04p_F^2/m$. The factor of three belongs to the three-dimensional angular average, not to a universal definition of effective mass.

A Landau quasiparticle is a fermionic excitation connected continuously to a Fermi gas particle when interactions preserve the phase. Near an isotropic Fermi surface, its energy relative to the chemical potential is ${\xi }_p={ϵ}_p-\mu \simeq v_F(p-p_F)$, with Fermi momentum $p_F$ and Fermi velocity $v_F$.^[ref]

The Pauli exclusion principle restricts decay. For a three-dimensional normal Fermi liquid with regular short-range interactions, a representative low-energy lifetime model isHere $E_{\ast }$ is an energy scale and $A$ is dimensionless; sharpness requires $\hslash /{\tau }_p\ll |{\xi }_p|$ at zero temperature.^[ref]

For an original example, stipulate $A=1$ and $T=0$. At $|{\xi }_p|/E_{\ast }=0.02$, the decay energy is $0.0004E_{\ast }$, just $2\%$ of the excitation energy; halving $|{\xi }_p|$ reduces that ratio to $1\%$. Approaching the Fermi surface improves the particle description even though the microscopic interaction has not weakened. This quadratic illustration is three-dimensional; low-dimensional corrections require separate analysis.

A normal Fermi liquid has interacting fermions whose low-energy excitations can be followed continuously from those of a Fermi gas, provided no phase transition destroys that correspondence. A Landau quasiparticle is a particle together with the surrounding many-body response. It has the appropriate charge and spin, a renormalized dispersion and a long lifetime close to the Fermi surface. Interactions remain in both the quasiparticle parameters and the interactions between quasiparticles; they have not been discarded.

Near an isotropic Fermi surface,The last relation defines the local isotropic Fermi-liquid effective mass. The Landau quasiparticle energy functional expresses how changing occupations shifts other quasiparticle energies:with normalization of the sum absorbed into $f$. Its angular components give Landau parameters describing collective density and spin response.

The Pauli exclusion principle severely restricts the available final states for low-energy collisions. In an ordinary three-dimensional Fermi liquid with regular interactions,At zero temperature the decay width decreases faster than $|\xi |$, making quasiparticles increasingly sharp near the Fermi surface. Their coherent spectral weight is the quasiparticle residue $0<Z\le 1$, and the true electron momentum distribution has a discontinuity of height $Z$ at zero temperature. The rest of the spectral weight is incoherent. The quasiparticle occupation itself still follows a Fermi-Dirac distribution.

The low-temperature electronic heat capacity is linear, $C_V/\mathcal{V}=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$, where $N^{\ast }(0)$ is the spin-summed quasiparticle density of states. In an isotropic spin-one-half model, the uniform spin susceptibility and compressibility take the formsThus enhanced effective mass and residual interactions can separately change observable responses. The familiar $T^2$ electrical resistivity requires a mechanism relaxing electrical current, such as allowed Umklapp processes; collisions in a perfectly clean Galilean-invariant continuum conserve total momentum and do not alone produce a finite dc resistivity.

Zero sound is a collisionless collective density oscillation sustained by the quasiparticle interaction; ordinary first sound occurs when collisions establish local equilibrium during a cycle. Metals and normal liquid helium-3 supply important examples. The description fails in phases with no sufficiently sharp low-energy quasiparticles, such as generic interacting one-dimensional systems, and must be modified after pairing or another ordering transition.

For a uniform continuum gas with pair interactions depending on relative positions, a Galilean boost changes every particle momentum by $p_0\hat{\mathbf{x}}$ and leaves the interaction energy unchanged. Write ${\hat{P}}_x={\sum }_j{\hat{p}}_{j,x}$. The transformed many-body Hamiltonian operator isThe resting ground state has $⟨{\hat{P}}_x⟩=0$, so the Galilean boost energy of an interacting gas isThe mass here is the bare particle mass, because this is an exact transformation of the microscopic kinetic energy. Substituting the Fermi-liquid effective mass $m^{\ast }$ would omit the change in interaction energy encoded by the Landau quasiparticle energy functional. This conclusion requires the Principle of Galilean relativity; a gas in a lattice or a fixed external potential energy landscape need not obey it.