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A real antisymmetric generator of the connected rotation group isChoose $D_{\mu }\mathbf{\varphi }=({\partial }_{\mu }-g{A}_{\mu }T)\mathbf{\varphi }$. Under $\mathbf{\varphi }\mapsto R(\alpha (x))\mathbf{\varphi }$ and $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha /g$, the gauge covariant derivative transforms in the same way as the field. Include the canonical gauge kinetic term $-F_{\mu \nu }{F}^{\mu \nu }/4$ so the vector's mass normalization is specified.

In unitary gauge, $\mathbf{\varphi }=(v+h,0{)}^T$. The relevant Lagrangian density isReading the quadratic vector term as $m_A^2{A}^2/2$ givesThe angular scalar supplies the longitudinal polarization of the massive vector; it is not a separate physical massless particle after gauging. The physical scalar still has $m_h^2=8\lambda v^2$.

The interactions among the physical fields areThus the complete physical vertex list is $hhh$, $hhhh$, $AAh$, and $AAhh$. An Abelian gauge kinetic term generates no $AAA$ or $AAAA$ self-interaction. Gauge-dependent Goldstone vertices in another gauge do not add physical particle species to this list. These statements concern the broken phase $v\ne 0$; when $v=0$ the unitary-gauge broken-phase parametrization is unavailable.

Put $\beta =(k_{B}T{)}^{-1}$ and label the two spin values in the order $+1,-1$. Assign each bond its Boltzmann weight,The symmetry of the bond energy makes $V$ a real symmetric matrix. For a fixed configuration the product of successive matrix entries is its Boltzmann weight. Summing over all spins and identifying the last index with the first therefore giveswhereThis replaces a sum over $2^L$ configurations by the powers of a two-by-two matrix. For finite bond energies at $T>0$, every matrix entry is positive, so the Perron–Frobenius theorem gives ${\lambda }_{+}>0$ and $|{\lambda }_{-}|<{\lambda }_{+}$. Hence the Helmholtz free energy per spin tends to $-k_{B}T\log {\lambda }_{+}$. A diagonal local observable $O$ can be inserted at one site: $⟨O_1⟩=\mathrm{tr}(O{V}^L)/Z_L$. Its limiting average is the expectation in the normalized eigenvector of ${\lambda }_{+}$.

The sum needs a stated interpretation. The original equation leaves the second bond outside explicit parentheses, with its index then formally free. The natural translation-invariant, equal-bond interpretation used here is the energy $-J{\sum }_j{\tau }_j({\sigma }_j+{\sigma }_{j+1})$.

Let $K=\beta J$. Sum over each decorating spin before summing over the remaining spins. The decimation of an alternating mixed-spin chain gives the effective transfer matrix:No on-site entropy has been omitted: the term $1$ is the weight of the zero decorating spin. The normalized eigenvectors are $v_{\pm }=(1,\pm 1{)}^{\mathsf{T}}/\sqrt{2}$, with eigenvaluesThere are $L$ transfer steps but $2L$ physical sites, soThe free energy per two-site unit cell is twice this value. At $J=0$, the formula gives $Z_L=6^L$, as required by two states on one sublattice and three on the other. As $T\to 0$, the free energy per physical site tends to $-J$, the energy of an aligned chain with two satisfied bonds per unit cell. These checks also fix the factor of two in the site count.

Separation of variables in a central potential writes a wavefunction as $R_{nl}(r)Y_l^m(\theta ,\varphi )$. With normalized spherical harmonics the radial factor satisfies ${\int }_0^{\infty }{r}^2|R_{nl}{|}^{2}dr=1$. The reduced radial function is $u=rR$, with normalization $\int |u{|}^{2}dr=1$; the two functions have different behavior at the origin. For a regular Coulomb eigenstate, $R_{nl}\propto r^l$ near the origin.