Sigma model

Taking ${\displaystyle g_{ij}=\delta _{ij}}$ the Kronecker delta, i.e. the scalar dot product in Euclidean space, one gets the ${\displaystyle O(n)}$ non-linear sigma model. That is, write ${\displaystyle \phi ={\hat {u}}}$ to be the unit vector in ${\displaystyle \mathbb {R} ^{n+1}}$, so that ${\displaystyle {\hat {u}}\cdot {\hat {u}}=1}$, with ${\displaystyle \cdot }$ the ordinary Euclidean dot product. Then ${\displaystyle {\hat {u}}\in S^{n}}$ the ${\displaystyle n}$-sphere, the isometries of which are the rotation group ${\displaystyle O(n)}$. The Lagrangian can then be written as

${\displaystyle {\mathcal {L}}={\frac {1}{2}}\nabla _{\mu }{\hat {u}}\cdot \nabla _{\mu }{\hat {u}}}$

For ${\displaystyle n=3}$, this is the continuum limit of the isotropic ferromagnet on a lattice, i.e. of the classical Heisenberg model. For ${\displaystyle n=2}$, this is the continuum limit of the classical XY model. See also the n-vector model and the Potts model for reviews of the lattice model equivalents. The continuum limit is taken by writing

${\displaystyle \delta _{h}[{\hat {u}}](i,j)={\frac {{\hat {u}}_{i}-{\hat {u}}_{j}}{h}}}$

as the finite difference on neighboring lattice locations ${\displaystyle i,j.}$ Then ${\displaystyle \delta _{h}[{\hat {u}}]\to \partial _{\mu }{\hat {u}}}$ in the limit ${\displaystyle h\to 0}$, and ${\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{j}\to \partial _{\mu }{\hat {u}}\cdot \partial _{\mu }{\hat {u}}}$ after dropping the constant terms ${\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{i}=1}$ (the "bulk magnetization").

This follows directly from the expression

${\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \phi ,\Delta \phi \rangle \!\rangle }$

given above. Taking ${\displaystyle \Phi =\mathbb {C} }$, the function ${\displaystyle \phi :M\to \mathbb {C} }$ can be interpreted as a wave function, and its Laplacian the kinetic energy of that wave function. The ${\displaystyle \langle \!\langle \cdot ,\cdot \rangle \!\rangle }$ is just some geometric machinery reminding one to integrate over all space. The corresponding quantum mechanical notation is ${\displaystyle \phi =|\psi \rangle .}$ In flat space, the Laplacian is conventionally written as ${\displaystyle \Delta =\nabla ^{2}}$. Assembling all these pieces together, the sigma model action is equivalent to

${\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\langle \psi |\nabla ^{2}|\psi \rangle dx^{m}={\frac {1}{2}}\int _{M}\psi ^{\dagger }(x)\nabla ^{2}\psi (x)dx^{m}}$

which is just the grand-total kinetic energy of the wave-function ${\displaystyle \psi (x)}$, up to a factor of ${\displaystyle \hbar /m}$. To conclude, the classical sigma model on ${\displaystyle \mathbb {C} }$ can be interpreted as the quantum mechanics of a free, non-interacting quantum particle. Obviously, adding a term of ${\displaystyle V(\phi )}$ to the Lagrangian results in the quantum mechanics of a wave-function in a potential. Taking ${\displaystyle \Phi =\mathbb {C} ^{n}}$ is not enough to describe the ${\displaystyle n}$-particle system, in that ${\displaystyle n}$ particles require ${\displaystyle n}$ distinct coordinates, which are not provided by the base manifold. This can be solved by taking ${\displaystyle n}$ copies of the base manifold.

It is very well-known that the geodesic structure of a Riemannian manifold is described by the Hamilton–Jacobi equations.[2] In thumbnail form, the construction is as follows. Both ${\displaystyle M}$ and ${\displaystyle \Phi }$ are Riemannian manifolds; the below is written for ${\displaystyle \Phi }$, the same can be done for ${\displaystyle M}$. The cotangent bundle ${\displaystyle T^{*}\Phi }$, supplied with coordinate charts, can always be locally trivialized, i.e.

${\displaystyle \left.T^{*}\Phi \right|_{U}\cong U\times \mathbb {R} ^{n}}$

The trivialization supplies canonical coordinates ${\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})}$ on the cotangent bundle. Given the metric tensor ${\displaystyle g_{ij}}$ on ${\displaystyle \Phi }$, define the Hamiltonian function

${\displaystyle H(q,p)={\frac {1}{2}}g^{ij}(q)p_{i}p_{j}}$

where, as always, one is careful to note that the inverse of the mertric is used in this definition: ${\displaystyle g^{ij}g_{jk}=\delta _{k}^{i}.}$ Famously, the geodesic flow on ${\displaystyle \Phi }$ is given by the Hamilton–Jacobi equations

${\displaystyle {\dot {q}}^{i}={\frac {\partial H}{\partial p_{i}}}\quad }$ and ${\displaystyle \quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q^{i}}}}$

The geodesic flow is the Hamiltonian flow; the solutions to the above are the geodesics of the manifold. Note, incidentally, that ${\displaystyle dH/dt=0}$ along geodesics; the time parameter ${\displaystyle t}$ is the distance along the geodesic.

The sigma model takes the momenta in the two manifolds ${\displaystyle T^{*}\Phi }$ and ${\displaystyle T^{*}M}$ and solders them together, in that ${\displaystyle \mathrm {d} \phi }$ is a solder form. In this sense, the interpretation of the sigma model as an energy functional is not surprising; it is in fact the gluing together of two energy functionals. Caution: the precise definition of a solder form requires it to be an isomorphism; this can only happen if ${\displaystyle M}$ and ${\displaystyle \Phi }$ have the same real dimension. Furthermore, the conventional definition of a solder form takes ${\displaystyle \Phi }$ to be a Lie group. Both conditions are satisfied in various applications.

For the special case of ${\displaystyle \Phi }$ being a Lie group, the ${\displaystyle g_{ij}}$ is the metric tensor on the Lie group, formally called the Cartan tensor or the Killing form. The Lagrangian can then be written as the pullback of the Killing form. Note that the Killing form can be written as a trace over two matrices from the corresponding Lie algebra; thus, the Lagrangian can also be written in a form involving the trace. With slight re-arrangements, it can also be written as the pullback of the Maurer-Cartan form.

A common variation of the sigma model is to present it on a symmetric space. The prototypical example is the chiral model, which takes the product

${\displaystyle G=SU(N)\times SU(N)}$

of the "left" and "right" chiral fields, and then constructs the sigma model on the "diagonal"

${\displaystyle \Phi ={\frac {SU(N)\times SU(N)}{SU(N)}}}$

Such a quotient space is a symmetric space, and so one can generically take ${\displaystyle \Phi =G/H}$ where ${\displaystyle H\subset G}$ is the maximal subgroup of ${\displaystyle G}$ that is invariant under the Cartan involution. The Lagrangian is still written exactly as the above, either in terms of the pullback of the metric on ${\displaystyle G}$ to a metric on ${\displaystyle G/H}$ or as a pullback of the Maurer-Cartan form.

In physics, the most common and conventional statement of the sigma model begins with the definition

${\displaystyle L_{\mu }=\pi _{\mathfrak {m}}\circ \left(g^{-1}\partial _{\mu }g\right)}$

Here, the ${\displaystyle g^{-1}\partial _{\mu }g}$ is the pullback of the Maurer-Cartan form, for ${\displaystyle g\in G}$, onto the spacetime manifold. The ${\displaystyle \pi _{\mathfrak {m}}}$ is a projection onto the odd-parity piece of the Cartan involution. That is, given the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of ${\displaystyle G}$, the involution decompses the space into odd and even parity components ${\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}$ corresponding to the two eigenstates of the involution. The sigma model Lagrangian can then be written as

${\displaystyle {\mathcal {L}}={\frac {1}{2}}\mathrm {tr} \left(L_{\mu }L^{\mu }\right)}$

This is instantly recognizable as the first term of the Skyrme model.

The equivalent metric form of this is to write a group element ${\displaystyle g\in G}$ as the geodesic ${\displaystyle g=\exp(\theta ^{i}T_{i})}$ of an element ${\displaystyle \theta ^{i}T_{i}\in {\mathfrak {g}}}$ of the Lie algebra ${\displaystyle {\mathfrak {g}}}$. The ${\displaystyle [T_{i},T_{j}]={f_{ij}}^{k}T_{k}}$ are the basis elements for the Lie algebra; the ${\displaystyle {f_{ij}}^{k}}$ are the structure constants of ${\displaystyle {\mathfrak {g}}}$.

Plugging this directly into the above and applying the infinitesimal form of the Baker–Campbell–Hausdorff formula promptly leads to the equivalent expression

${\displaystyle {\mathcal {L}}={\frac {1}{2}}g_{ij}(\phi )\;\mathrm {d} \phi _{i}\wedge {*\mathrm {d} \phi _{j}}={\frac {1}{2}}\;{W_{i}}^{m}{W^{n}}_{j}\;\;\mathrm {d} \phi _{i}\wedge {*\mathrm {d} \phi _{j}}\;\;\mathrm {tr} (T_{m}T_{n})}$

where ${\displaystyle \mathrm {tr} (T_{m}T_{n})}$ is now obviously (proportional to) the Killing form, and the ${\displaystyle {W_{i}}^{m}}$ are the vielbeins that express the "curved" metric ${\displaystyle g_{ij}}$ in terms of the "flat" metric ${\displaystyle \mathrm {tr} (T_{m}T_{n})}$. The article on the Baker–Campbell–Hausdorff formula provides an explicit expression for the vielbeins. They can be written as

${\displaystyle W=\sum _{n=0}^{\infty }{\frac {(-1)^{n}M^{n}}{(n+1)!}}=(I-e^{-M})M^{-1}}$

where ${\displaystyle M}$ is a matrix whose matrix elements are ${\displaystyle {M_{j}}^{k}=\theta ^{i}{f_{ij}}^{k}}$.

For the sigma model on a symmetric space, as opposed to a Lie group, the ${\displaystyle T_{i}}$ are limited to span the subspace ${\displaystyle {\mathfrak {m}}}$ instead of all of ${\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}$. The Lie commutator on ${\displaystyle {\mathfrak {m}}}$ will not be within ${\displaystyle {\mathfrak {m}}}$; indeed, one has ${\displaystyle [{\mathfrak {m}},{\mathfrak {m}}]\subset {\mathfrak {h}}}$ and so a projection is still needed.