Symmetry Group in Two Continuous Dimensional Space

Continuous Symmetry

Topological Defects and Their Homotopy Classification

T.W.B. Kibble , in Encyclopedia of Mathematical Physics, 2006

Examples

The simplest continuous symmetry is the U(1) phase symmetry $\hat{\varphi }\mapsto \hat{\varphi }{\text{e}}^{\text{i}\alpha }$ of a complex field. In a weakly interacting Bose gas, below the Bose–Einstein condensation temperature, or in superfluid helium-4, a macroscopic fraction of the atoms occupies a single quantum state, and $\hat{\varphi }$ acquires a nonzero expectation value, $〈\hat{\varphi }〉=\varphi$ , whose phase is arbitrary, so the symmetry is completely broken to H = 1. Thus, $\mathcal{M}=S^1$ we have a circle of equivalent degenerate ground states. (This corresponds to spontaneous breaking of the particle-number symmetry. It is possible to describe the system in a U(1)-invariant way, by projecting out a state of definite particle number, a uniform superposition of all the states in $\mathcal{M}$ , but it is generally less convenient to do so.) In this case, the only nontrivial homotopy group is ${\pi }_1\left(\mathcal{M}\right)=\mathbb{Z}$ , so the only defects are linear defects classified by a winding number $n\in \mathbb{Z}$ . The defects with n = ±1 are stable vortices. Those with |n| > 1 are in general unstable and tend to break up into |n| single-quantum vortices.

Low-temperature superconductors also have a U(1) symmetry, although there are important differences. This is not a global symmetry but a local, gauge symmetry, with coupling to the electromagnetic field. Moreover, it is not single atoms that condense but Cooper pairs, pairs of electrons of equal and opposite momentum and spin. These systems too exhibit linear defects, magnetic flux tubes carrying a magnetic flux 4πnħ/e.

A less trivial example is a nematic liquid crystal. These materials are composed of rod-shaped molecules that tend, at low temperatures, to line up parallel to one another. The nematic state is characterized by a preferred orientation, described by a unit vector n , the director. (Note that n and − n are physically equivalent.) There is long-range orientational order, with molecules preferentially lining up parallel to n , but unlike a solid crystal there is no long-range translational order – the molecules move freely past each other as in a normal liquid.

A convenient order parameter here is the mean mass quadrupole tensor Φ of a molecule. In the nematic state, Φ is proportional to (3 nn − 1); for example, if n = (0,0,1), then Φ is diagonal with diagonal elements proportional to (−1, −1, 2). In this case, the symmetry group is SO(3) (or, more precisely, O(3); but the inversion symmetry is not broken, so we can restrict our attention to the connected part of the group). The subgroup H that leaves this Φ invariant is a semidirect product, $H=\text{SO}\left(2\right)⋉{\mathbb{Z}}_2$ (isomorphic to O(2)), composed of rotations about the z-axis and rotations through π about axes in the x–y plane. (If we enlarge G to its simply connected covering group $\tilde{G}=\text{SU}\left(2\right)$ , then H becomes $\tilde{H}=\left[\text{U}\left(1\right)⋉{\mathbb{Z}}_4\right]/{\mathbb{Z}}_2$ , where U(1) is generated as before by J_z . The essential difference is that the square of any of the elements in the disconnected piece of $\tilde{H}$ is not now the identity but the element e^{2πiJ _z} = −1 ∈ U(1).) The manifold $\mathcal{M}$ of degenerate ground states in this case is the projective space $\mathbb{R}{P}^2$ (obtained by identifying opposite points of S ²).

Since $\tilde{H}$ has disconnected pieces, we have ${\pi }_1\left(\mathcal{M}\right)={\pi }_0\left(\tilde{H}\right)={\mathbb{Z}}_2$ . Thus, there can be topologically stable linear defects, here called disclination lines, around which the director n rotates by π (see Figure 6 ). The fact that these defects are classified by ${\mathbb{Z}}_2$ rather than $\mathbb{Z}$ means that a line around which n rotates by 2π is topologically trivial; indeed, n can be smoothly rotated near the line to run parallel to it, leaving a configuration with no defect.

Figure 6. Orientation of molecules around a disclination line.

There are also point defects; since ${\pi }_2\left(\mathcal{M}\right)={\pi }_1\left(\tilde{H}\right)=\mathbb{Z}$ , they are labeled by an integer winding number n. In a defect with n = 1, the vector n points radially outwards all round the defect position.

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Einstein Equations: Exact Solutions

Jiří Bičák , in Encyclopedia of Mathematical Physics, 2006

Classification According to Symmetries

Most of the available solutions have some exact continuous symmetries which preserve the metric. The corresponding group of motions is characterized by the number and properties of its Killing vectorsξ ^α satisfying the Killing equation(£ _ξ g) _αβ = ξ _α;β + ξ _β;α = 0 (£ is the Lie derivative) and by the nature (spacelike, timelike, or null) of the group orbits. For example, axisymmetric, stationary fields possess two commuting Killing vectors, of which one is timelike. Orbits of the axial Killing vector are closed spacelike curves of finite length, which vanishes at the axis of symmetry. In cylindrical symmetry, there exist two spacelike commuting Killing vectors. In both cases, the vectors generate a two-dimensional abelian group. The two-dimensional group orbits are timelike in the stationary case and spacelike in the cylindrical symmetry.

If a timelike ξ ^α is hypersurface orthogonal, ξ _α = λΦ_,α for some scalar functions λ,Φ, the spacetime is "static." In coordinates with ξ = ∂ _t , the metric is

[6] $g=-{\text{e}}^{\text{2}U}d{t}^2+{\text{e}}^{-2U}{\gamma }_{ik}d{x}^{i}d{x}^k$

where U,γ _ik do not depend on t. In vacuum, U satisfies the potential equation $U_{:a}^{:a}=0$ , the covariant derivatives (denoted by:) are with respect to the three-dimensional metric γ _ik . A classical result of Lichnerowicz states that if the vacuum metric is smooth everywhere and U → 0 at infinity, the spacetime is flat (for refinements, see Anderson (2000)).

In cosmology, we are interested in groups whose regions of transitivity (points can be carried into one another by symmetry operations) are three-dimensional spacelike hypersurfaces (homogeneous but anisotropic models of the universe). The three-dimensional simply transitive groups G ₃ were classified by Bianchi in 1897 according to the possible distinct sets of structure constants but their importance in cosmology was discovered only in the 1950s. There are nine types: Bianchi I to Bianchi IX models. The line element of the Bianchi universes can be expressed in the form

[7] $g=-d{t}^2+g_{ab}(t){\omega }^a{\omega }^b$

where the time-independent 1-forms ${\omega }^a=E_{\alpha }^{a}d{x}^{\alpha }$ satisfy the relations $d{\omega }^a=-\left(1/2\right)C_{bc}^a{\omega }^b\wedge {\omega }^c$ , d is the exterior derivative and $C_{bc}^a$ are the structure constants (see Cosmology: Mathematical Aspects for more details).

The standard Friedmann–Lemaître–Robertson–Walker (FLRW) models admit in addition an isotropy group SO(3) at each point. They can be represented by the metric

[8] $g=-d{t}^2+{[a(t)]}^2\left(\frac{d{r}^2}{1-k{r}^2}+r^2(d{\theta }^2+{sin}^2\theta d{\phi }^2)\right)$

in which a(t), the "expansion factor," is determined by matter via EEs, the curvature index k = −1,0, +1, the three-dimensional spaces t = const. have a constant curvature K = k/a ²;r ∈ [0,1] for closed (k = +1) universe, r ∈ [0,∞) in open (k = 0, −1) universes (for another description (see Cosmology: Mathematical Aspects).

There are four-dimensional spacetimes of constant curvature solving EEs [2] with T _μν = 0: the Minkowski, de Sitter, and anti-de Sitter spacetimes. They admit the same number [10] of independent Killing vectors, but interpretations of the corresponding symmetries differ for each spacetime.

If ξ ^α satisfies £ _ξ g _αβ = 2Φg _αβ ,Φ = const., it is called a homothetic (Killing) vector. Solutions with proper homothetic motions, Φ ≠ 0, are "self-similar." They cannot in general be asymptotically flat or spatially compact but can represent asymptotic states of more general solutions. In Stephani et al. (2003), a summary of solutions with proper homotheties is given; their role in cosmology is analyzed by Wainwright and Ellis (eds.) (1997); for mathematical aspects of symmetries in general relativity, see Hall (2004).

There are other schemes for invariant classification of exact solutions (reviewed in Stephani et al. (2003)): the algebraic classification of the Ricci tensor and energy–momentum tensor of matter; the existence and properties of preferred vector fields and corresponding congruences; local isometric embeddings into flat pseudo-Euclidean spaces, etc.

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Phase Transitions: Exact (or Almost Exact) Results for Various Models

R.K. Pathria , Paul D. Beale , in Statistical Mechanics (Third Edition), 2011

13.3 The n-vector models in one dimension

We now consider a generalization of the Ising chain in which the spin variable ∑ _i is an n-dimensional vector of magnitude unity, whose components can vary continuously over the range -1 to +1; in contrast, the Ising spin ∑ _i could have only a discrete value, +1 or -1. We shall see that the vector models (with n ≥ 2), while differing quantitatively from one another, differ rather qualitatively from the scalar models (for which n = 1). While some of these qualitative differences will show up in the present study, more will become evident in higher dimensions. Here we follow a treatment due to Stanley (1969a,b) who first solved this problem for general n.

Once again we employ an open chain composed of N spins constituting (N - 1) nearest-neighbor pairs. The Hamiltonian of the system, in zero field, is given by

(1) $H_N\left\{{\sigma }_i\right\}=-{\displaystyle \sum _{i=1}^{N-1}{J}_i{\sigma }_iċ{\sigma }_{i+1}}.$

We assume our spins to be classical, so we do not have to worry about the commutation properties of their components. And since the components ∑ _iα (α = 1, …, n) of each spin vector ∑ _i are now continuous variables, the partition function of the system will involve integrations, rather than summations, over these variables. Associating equal a priori probabilities with solid angles of equal magnitude in the n-dimensional spin-vector space, we may write

(2) $Q_N={\displaystyle \int \frac{d{\Omega }_1}{\Omega (n)}}\cdots \frac{d{\Omega }_N}{\Omega (n)}{\displaystyle \prod _{i=1}^{N-1}{e}^{\beta J_i{\sigma }_i.{\sigma }_i+1}},$

where Ω(n) is the total solid angle in an n-dimensional space; see equation (7b) of Appendix C, which gives

(3) $\Omega (n)=2{\pi }^{n/2}/\Gamma (n/2).$

We first carry out integration over ∑ _N , keeping the other ∑ _i fixed. The relevant integral to do is

(4) $\frac{1}{\Omega (n)}{\displaystyle \int e^{\beta J_{N-1}{\sigma }_{N-1}.{\sigma }_N}d{\Omega }_N}.$

For ∑ _N we employ spherical polar coordinates, with polar axis in the direction of ∑ _N–1, while for dΩ _N we use expression (9) of Appendix C. Integration over angles other than the polar angle θ yields a factor of

(5) $2{\pi }^{(n-1)/2}/\Gamma \{(n-1)/2\}.$

The integral over the polar angle is

(6) ${\displaystyle \underset{0}{\overset{\pi }{\int }}{e}^{\beta J_{N-1}\cos \theta }{\sin }^{n-2}\theta d\theta }=\frac{{\pi }^{1/2}\Gamma \{(n-1)/2\}}{{(\frac{1}{2}\beta J_{N-1})}^{(n-2)/2}}{I}_{(n-2)/2}(\beta J_{N-1}),$

where I_μ (x) is a modified Bessel function; see Abramowitz and Stegun (1964), formula 9.6.18. Combining (3), (5), and (6), we obtain for (4) the expression

(7) $\frac{\Gamma (n/2}{{(\frac{1}{2}\beta J_{N-1})}^{(n-2)/2}}{I}_{(n-2)/2}(\beta J_{N-1}),$

regardless of the direction of ∑ _N–1. By iteration, we get

(8) $Q_N={\displaystyle \prod _{i=1}^{N-1}\frac{\Gamma (n/2}{{(\frac{1}{2}\beta J_i)}^{(n-2)/2}}{I}_{(n-2)/2}(\beta J_i);}$

the last integration, over dΩ₁, gave simply a factor of unity.

Expression (8) is valid for all n — including n = 1, for which it gives: Q_N = Π _i cosh (βJ_i ). This last result differs from expression (13.2.27) by a factor of 2 ^N ; the reason for this difference lies in the fact that the Q_N of the present study is normalized to go to unity as the βJ_i go to zero [see equation (2)] whereas the Q_N of the preceding section, being a sum over 2 ^N discrete states [see equation (13.2.24)] goes to 2 ^N instead. This difference is important in the evaluation of the entropy of the system; it is of no consequence for the calculations that follow.

First of all we observe that the partition function Q_N is analytic at all β — except possibly at β = ∞ where the singularity of the problem is expected to lie. Thus, no long-range order is expected to appear at any finite temperature T — except at T = 0 where, of course, perfect order is supposed to prevail. In view of this, the correlation function for the nearest-neighbor pair (∑ _k , ∑ _k+1) is simply $\overline{{\sigma }_kċ{\sigma }_{k+1}}$ and is given by, see equations (2) and (8),

(9) $g_k(n.n.)=\frac{1}{Q_N}\left(\frac{1}{\beta }\frac{\partial }{\partial J_k}\right)Q_N=\frac{I_{n/2}(\beta J_k)}{I_{n-2/2}(\beta J_k)}.$

The internal energy of the system turns out to be

(10) $U_0\equiv -\frac{\partial }{\partial \beta }(\text{In}{Q}_N)=-{\displaystyle \sum _{i=1}^{N-1}{J}_i}\frac{I_{n/2}(\beta J_i)}{I_{n-2/2}(\beta J_i)};$

not surprisingly, U ₀ is simply a sum of the expectation values of the nearest-neighbor interaction terms -J_i ∑ _i · ∑ _i+1, which is identical to a sum of the quantities –J_ig_i (n.n.) over all nearest-neighbor pairs in the system.

The calculation of g_k (r) is somewhat tricky because of the vector character of the spins, but things are simplified by the fact that we are dealing with a one-dimensional system only. Let us consider the trio of spins ∑ _k , ∑ _k+1 and ∑_k+2, and suppose for a moment that our spins are three-dimensional vectors; our aim is to evaluate $\overline{{\sigma }_kċ{\sigma }_{k+2}}$ . We choose spherical polar coordinates with polar axis in the direction of ∑ _k+1; let the direction of ∑ _k be defined by the angles (θ₀, φ₀) and that of ∑ _k+2 by (θ₂, φ₂). Then

(11) $\overline{{\sigma }_kċ{\sigma }_{k+2}}=\overline{\cos \theta (k,k+2)}=\overline{\cos {\theta }_0\cos {\theta }_2+\sin {\theta }_0\sin {\theta }_2\cos ({\varphi }_0-{\varphi }_2)}.$

Now, with ∑ _k+1 fixed, spins ∑ _k and ∑ _k+2 will orient themselves independently of one another because, apart from ∑ _k+1, there is no other channel of interaction between them. Thus, the pairs of angles (θ₀, φ₀) and (θ₂, φ₂) vary independently of one another; this makes $\overline{\cos ({\varphi }_0-{\varphi }_2)}=0$ and $\overline{\cos {\theta }_0\cos {\theta }_2}=\overline{\cos ({\theta }_0)}\overline{\cos ({\theta }_2)}$ . It follows that

(12) $\overline{{\sigma }_kċ{\sigma }_{k+2}}=\overline{{\sigma }_kċ{\sigma }_{k+1}}\overline{{\sigma }_{k+1}ċ{\sigma }_{k+2}}.$

Extending this argument to general n and to a segment of length r, we get

(13) $g_k(r)={\displaystyle \prod _{i=k}^{k+r-1}{g}_i(\text{n}\text{.n}\text{.})}={\displaystyle \prod _{i=k}^{k+r-1}{I}_{n/2}(\beta J_i)}/I_{(n-2)/2}(\beta J_i).$

With a common J, equations (9), (10), and (13) take the form

(14) $g(\text{n}\text{.n}\text{.})=I_{n/2}(\beta J)/I_{(n-2)/2}(\beta J),$

(15) $U_0=-(N-1)J{I}_{n/2}(\beta J)/I_{(n-2)/2}(\beta J)$

and

(16) $g(r)={\{I_{n/2}(\beta J)/I_{(n-2)/2}(\beta J)\}}^r.$

The last result here may be written in the standard form e^-r/ξ , with

(17) $\xi ={[\text{In{}{I}_{(n-2)/2}(\beta J)/I_{n/2}(\beta J)\}]}^{-1}.$

For n = 1, we have: I ₁/₂(x)/I _–1/₂(x) = tanh x; the results of the preceding section are then correctly recovered.

For a study of the low-temperature behavior, where βJ ≫ 1, we invoke the asymptotic expansion

(18) $I_{\mu }(x)=\frac{e^x}{\surd (2\pi x)}\left[1-\frac{4{\mu }^2-1}{8x}+\cdots \right](x\gg 1),$

with the result that

(14a) $g(\text{n}\text{.n}\text{.})\approx 1-\frac{n-1}{2\beta J},$

(15a) $U_0\approx -(N-1)J\left[1-\frac{n-1}{2\beta J}\right]$

and

(17a) $\xi \approx \frac{2\beta J}{n-1}\sim T^{-1}.$

Clearly, the foregoing results hold only for n ≥ 2; for n = 1, the asymptotic expansion (18) is no good because it yields the same result for $\mu =\frac{1}{2}$ as for $\mu =-\frac{1}{2}$ . In that case one is obliged to use the closed form result, g(n.n.) = tanh(βJ), which for βJ ≫ 1 gives

(14b) $g(\text{n}\text{.n}\text{.})\approx 1-2e^{-2\beta J}$

(15b) $U_0\approx -(N-1)J\left[1-2e^{-2\beta J}\right]$

and

(17b) $\xi \approx \frac{1}{2}{e}^{2\beta J},$

in complete agreement with the results of the preceding section. For completeness, we have for the low-temperature specific heat of the system

(19a) (19b) $C_0\approx (N-1)\{\begin{array}{ll}\frac{1}{2}(n-1)k\hfill & \text{for}n\ge 2\hfill \\ 4k{(\beta J)}^2{e}^{-2\beta J}\hfill & \text{for}n=1.\hfill \end{array}$

The most obvious distinction between one-dimensional models with continuous symmetry (n ≥ 2) and those with discrete symmetry (n = 1) is in relation to the nature of the singularity at T = 0. While in the case of the former it is a power-law singularity, with critical exponents ¹

(20) $\alpha =1,v=1,\eta =1,\text{and hence}\gamma \text{=1,}$

in the case of the latter it is an exponential singularity. Nevertheless, by introducing the temperature parameter t = e^-pβJ , see equation (13.2.17), we converted this exponential singularity in T into a power-law singularity in t, with

(21) $\alpha =2-2p\text{,}\gamma =v=2/p,\eta =1.$

However, the inherent arbitrariness in the choice of the number p left an ambiguity in the values of these exponents; we now see that by choosing p = 2 we can bring exponents (21) in line with (20).

Next we observe that the critical exponents (20) for n ≥ 2 turn out to be independent of n — a feature that seems peculiar to situations where T_c = 0. In higher dimensions, where T_c is finite, the critical exponents do vary with n; for details, see Section 13.7. In any case, the amplitudes always depend on n. In this connection we note that, since each of the N spins comprising the system has n components, the total number of degrees of freedom in this problem is Nn. It seems appropriate that the extensive quantities, such as U ₀ and C ₀, be divided by Nn, so that they are expressed as per degree of freedom. A look at equation (15a) then tells us that our parameter J must be of the form nJ′, so that in the thermodynamic limit

(22) $\frac{U_0}{N_n}\approx -J^{\prime }+\frac{n-1}{2n}kT$

and, accordingly,

(23) $\frac{C_0}{N_n}\approx \frac{n-1}{2n}k.$

Equation (17a) then becomes

(24) $\xi \approx \frac{2n}{n-1}\frac{J^{\prime }}{kT}.$

Note that the amplitudes appearing in equations (22) through (24) are such that the limit n → ∞ exists; this limit pertains to the so-called spherical model, which will be studied in Section 13.5.

Figure 13.6 shows the normalized energy u ₀(= U ₀ /NnJ′) as a function of temperature for several values of n, including the limiting case n = 1. We note that u ₀ (which, in the thermodynamic limit, is equal and opposite to the nearest-neighbor correlation function g(n.n.)) increases monotonically with n — implying that g(n.n.), and hence g(r), decrease monotonically as n increases. This is consistent with the fact that the correlation length ξ also decreases as n increases; see equation (24). The physical reason for this behavior is that an increase in the number of degrees of freedom available to each spin in the system effectively diminishes the degree of correlation among any two of them.

FIGURE 13.6. The normalized energy u₀(= U ₀ /NnJ′) of a one-dimensional chain as a function of the temperature parameter kT/J′ for several values of n (after Stanley, 1969a,b). Note that for this classical model, the slopes as T → 0 are given by the equipartition theorem for n - 1 degrees of freedom per spin.

Another feature emerges here that distinguishes vector models (n ≥ 2) from the scalar model (n = 1); this is related to the manner in which the quantity u ₀ approaches its ground-state value -1. While for n = 1, the approach is quite slow—leading to a vanishing specific heat, see equations (15b) and (19b) — for n ≥ 2, the approach is essentially linear in T, leading to a finite specific heat; see equations (15a) and (19a). This last result violates the third law of thermodynamics, according to which the specific heat of a real system must go to zero as T → 0. The resolution of this dilemma lies in the fact that the low-lying states of a system with continuous symmetry (n ≥ 2) are dominated by long-wavelength excitations, known as Goldstone modes, which in the case of a magnetic system assume the form of "spin waves," characterized by a particle-like spectrum: ω(k) ∼ k ². The very low-temperature behavior of the system is primarily governed by these modes, and the thermal energy associated with them is given by

(25) $U_{\text{thermal}}\sim {\displaystyle \int \frac{ћ\omega }{\exp (ћ\omega /k_{B}T)-1}{k}^{d-1}dk\sim T^{(d+2)/2}};$

this results in a specific heat∼ T^{d /2}, which indeed is consistent with the third law. For a general account of the Goldstone excitations, see Huang (1987); for their role as "spin waves" in a magnetic system, see Plischke and Bergersen (1989).

For further information on one-dimensional models, see Lieb and Mattis (1966) and Thompson (1972a,b).

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Introductory Article: Equilibrium Statistical Mechanics

G. Gallavotti , in Encyclopedia of Mathematical Physics, 2006

Appendix 3: An Infrared Inequality

The infrared inequalities stem from Bogoliubov's inequality. Consider as an example the problem of crystallization discussed in the section "Continuous symmetries: 'no d = 2 crystal' theorem". Let 〈·〉 denote average over a canonical equilibrium state with Hamiltonian

$H={\displaystyle \sum _{j=1}^N}\frac{{\mathit{p}}_j^2}{2}+U\left(\mathit{Q}\right)+\varepsilon W\left(\mathit{Q}\right)$

with given temperature and density parameters β,ρ,ρ = a ⁻³. Let $\left\{X,Y\right\}={\sum }_j\left({\partial }_{p_j}X{\partial }_{q_j}Y-{\partial }_{q_j}X{\partial }_{p_j}Y\right)$ be the Poisson bracket. Integration by parts, with periodic boundary conditions, yields

[75] $\begin{array}{l}〈A*\left\{C,H\right\}〉\equiv -\frac{\int A*\left\{C,{\text{e}}^{-\beta H}\right\}\text{d}\mathit{P}\text{d}\mathit{Q}}{\beta Z_{\text{c}}\left(\beta ,\rho ,N\right)}\hfill \\ \equiv -{\beta }^{-1}〈\left\{A*,C\right\}〉\hfill \end{array}$

as a general identity. The latter identity implies, for A = {C,H}, that

[76] $〈\left\{H,C\right\}*\left\{H,C\right\}〉=-{\beta }^{-1}〈\left\{C,\left\{H,C*\right\}\right\}〉$

Hence, the Schwartz inequality $〈A*A〉〈\left\{H,C\right\}*\left\{H,C\right\}〉\ge {\left|〈\left\{A*,C\right\}〉\right|}^2$ combined with the two relations in [75], [76] yields Bogoliubov's inequality:

[77] $〈A*A〉\ge {\beta }^{-1}\frac{{\left|〈\left\{A*,C\right\}〉\right|}^2}{〈\left\{C,\left\{C*,H\right\}\right\}〉}$

Let g, h be arbitrary complex (differentiable) functions and ∂ _j = ∂ _{q j}

[78] $A\left(\mathit{Q}\right){\displaystyle \stackrel{\text{def}}{=}}{\displaystyle \sum _{j=1}^N}g\left({\mathit{q}}_j\right),C\left(\mathit{P},\mathit{Q}\right){\displaystyle \stackrel{\text{def}}{=}}{\displaystyle \sum _{j=1}^N}{\mathit{p}}_{j}h\left({\mathit{q}}_j\right)$

Then $H=\sum \frac{1}{2}{\mathit{p}}_j^2+\mathbf{\Phi }\left({\mathit{q}}_1,\dots ,{\mathit{q}}_N\right)$ , if

$\mathbf{\Phi }\left({\mathit{q}}_1,\dots {\mathit{q}}_N\right)=\frac{1}{2}{\displaystyle \sum _{j\ne j\prime }}\phi \left(\left|{\mathit{q}}_j-{\mathit{q}}_{j\prime }\right|\right)+\varepsilon {\displaystyle \sum _j}W\left({\mathit{q}}_j\right)$

so that, via algebra,

$\left\{\mathit{C},H\right\}\equiv {\displaystyle \sum _j}\left(h_j{\partial }_j\mathbf{\Phi }-{\mathit{p}}_j\left({\mathit{p}}_j\cdot {\partial }_j\right)h_j\right)$

with $h_j{\displaystyle \stackrel{\text{def}}{=}}h\left({\mathit{q}}_j\right)$ . If h is real valued, 〈{C,{C*,H}}〉 becomes, again via algebra,

$〈{\displaystyle \sum _{\text{<mi mathvariant="italic" is="true">jj</mi>}\prime }}{h}_j{h}_{j\prime }{\partial }_j\cdot {\partial }_{j\prime }\mathbf{\Phi }\left(\mathit{Q}\right)〉+〈\varepsilon {\displaystyle \sum _j}{h}_j^2\Delta W\left({\mathit{q}}_j\right)+\frac{4}{\beta }{\displaystyle \sum _j}{\left({\partial }_j{h}_j\right)}^2〉$

(integrals on p _j just replace ${\mathit{p}}_j^2$ by 2β ⁻¹ and $〈{\left({\mathit{p}}_j\right)}_i{\left({\mathit{p}}_j\right)}_{i\prime }〉={\beta }^{-1}{\delta }_{i,i\prime }$ ). Therefore, the average 〈{C,{C*,H}}〉 becomes

[79] $〈\frac{1}{2}{\displaystyle \sum _{\text{<mi mathvariant="italic" is="true">jj</mi>}\prime }}{\left(h_j-h_{j\prime }\right)}^2\Delta \phi \left(\left|{\mathit{q}}_j-{\mathit{q}}_{j\prime }\right|\right)+\varepsilon {\displaystyle \sum _j}{h}_j^2\Delta W\left({\mathit{q}}_j\right)+4{\beta }^{-1}{\displaystyle \sum _j}{\left({\partial }_j{h}_j\right)}^2〉$

Choose $g\left(\mathit{q}\right)\equiv e^{-1\left(\mathit{\kappa }+\mathit{K}\right)\cdot \mathit{q}},h\left(\mathit{q}\right)=cos\mathit{q}\cdot \mathit{\kappa }$ and bound (h _j − h _j′)² by ${\mathit{\kappa }}^2{\left({\mathit{q}}_j-{\mathit{q}}_{j\prime }\right)}^2,{\left({\partial }_j{h}_j\right)}^2$ by κ ² and $h_j^2$ by 1. Hence [79] is bounded above by ND( κ ) with

[80] $D\left(\mathit{\kappa }\right){\displaystyle \stackrel{\text{def}}{=}}〈{\mathit{\kappa }}^2\left(4{\beta }^{-1}+\frac{1}{2N}{\displaystyle \sum _{j\ne j\prime }}{\left({\mathit{q}}_j-{\mathit{q}}_{j\prime }\right)}^2\left|\Delta \phi \left({\mathit{q}}_j-{\mathit{q}}_{j\prime }\right)\right|\right)+\varepsilon \frac{1}{N}{\displaystyle \sum _j}\left|\Delta W\left({\mathit{q}}_j\right)\right|〉$

This can be used to estimate the denominator in [77]. For the LHS remark that

$〈A*,A〉={\left|{\displaystyle \sum _{j=1}^N}{\text{e}}^{-\text{i}q\cdot \left(\mathit{\kappa }+\mathit{K}\right)}\right|}^2$

and

$\begin{array}{l}{\left|〈\left\{A*,C\right\}〉\right|}^2={\left|〈{\displaystyle \sum _j}{h}_j\partial g_j〉\right|}^2\hfill \\ ={\left|\mathit{K}+\mathit{\kappa }\right|}^2{N}^2{\left({\rho }_{\varepsilon }\left(\mathit{K}\right)+{\rho }_{\varepsilon }\left(\mathit{K}+2\mathit{\kappa }\right)\right)}^2\hfill \end{array}$

hence [77] becomes, after multiplying both sides by the auxiliary function γ( κ ) (assumed even and vanishing for | κ | > π/a) and summing over κ ,

[81] $\begin{array}{l}{D}_1{\displaystyle \stackrel{\text{def}}{=}}\frac{1}{N}{\displaystyle \sum _{\mathit{\kappa }}}\text{γ}\left(\mathit{\kappa }\right)〈\frac{1}{N}{\left|{\displaystyle \sum _{j=1}^N}{\text{e}}^{-\text{i}\left(\mathit{K}+\mathit{\kappa }\right)\cdot {\mathit{q}}_j}\right|}^2〉\hfill \\ \ge \frac{1}{N}{\displaystyle \sum _{\mathit{\kappa }}}\text{γ}\left(\mathit{\kappa }\right)\times \frac{{\left|\mathit{K}\right|}^2}{4\beta }\frac{{\left({\rho }_{\varepsilon }\left(\mathit{K}\right)+{\rho }_{\varepsilon }\left(\mathit{K}+2\mathit{\kappa }\right)\right)}^2}{D\left(\mathit{\kappa }\right)}\hfill \end{array}$

To apply [77] the averages in [80], [81] have to be bounded above: this is a technical point that is discussed here, as it illustrates a general method of using the results on the thermodynamic limits and their convexity properties to obtain estimates.

Note that $〈\left(1/N\right){\sum }_k\text{γ}\left(\mathit{k}\right){\text{d}}^d\mathit{k}{\left|{\sum }_{j=1}^N{\text{e}}^{-\text{i}\mathit{k}\cdot {\mathit{q}}_j}\right|}^2〉$ is identically ${\displaystyle \tilde{\phi }}\left(0\right)+\left(2/N\right)〈{\sum }_{j<j\prime }{\displaystyle \tilde{\phi }}\left({\mathit{q}}_j-{\mathit{q}}_{j\prime }\right)〉$ with ${\displaystyle \tilde{\phi }}\left(\mathit{q}\right){\displaystyle \stackrel{\text{def}}{=}}\left(1/N\right){\sum }_{\mathit{\kappa }}\text{γ}\left(\mathit{\kappa }\right){\text{e}}^{\text{i}\mathit{\kappa }\cdot \mathit{q}}$ .

Let ${\phi }_{\lambda ,\zeta }\left(\mathit{q}\right){\displaystyle \stackrel{\text{def}}{=}}\phi \left(\mathit{q}\right)+\lambda {\mathit{q}}^2\left|\Delta \phi \left(\mathit{q}\right)\right|+\eta {\displaystyle \tilde{\phi }}\left(\mathit{q}\right)$ and let $F_V\left(\lambda ,\eta ,\zeta \right){\displaystyle \stackrel{\text{def}}{=}}\left(1/N\right)log{Z}^{\text{c}}\left(\lambda ,\eta ,\zeta \right)$ with Z ^cthe partition function in the volume Ω computed with energy $U\prime ={\sum }_{\text{<mi mathvariant="italic" is="true">jj</mi>}\prime }{\phi }_{\lambda ,\zeta }\left({\mathit{q}}_j-{\mathit{q}}_{j\prime }\right)+\varepsilon {\sum }_{j}W\left({\mathit{q}}_j\right)+\eta \varepsilon \sum \left|\Delta W\left({\mathit{q}}_j\right)\right|$ . Then F _V (λ,η,ζ) is convex in λ,η and it is uniformly bounded above and below if |η|,|ɛ|,|ζ| ≤ 1 (say) and |λ| ≤ λ ₀: here λ ₀ > 0 exists if r ²|Δφ( r )| satisfies the assumption set at the beginning of the section "Continuous symmetries: 'no d = 2 crystal' theorem" and the density is smaller than a close packing (this is because the potential U′ will still satisfy conditions similar to [14] uniformly in |ɛ|,|η| < 1 and |λ| small enough).

Convexity and boundedness above and below in an interval imply bounds on the derivatives in the interior points, in this case on the derivatives of F _V with respect to λ,η,ζ at 0. The latter are identical to the averages in [80], [81]. In this way, the constants B ₁,B ₂,B ₀ such that D( κ ) ≤ κ ² B ₁ + ɛB ₂ and B ₀ > D ₁ are found.

For more details, the reader is referred to Mermin (1968).

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Non-Fermi Liquid Behavior in Heavy Fermion Systems

Pedro Schlottmann , in Handbook of Magnetic Materials, 2015

4.4 QCPs in 2D Models

To understand the origin of QCPs in 2D it useful to analyze the Mermin-Wagner-Hohenberg theorem (Hohenberg, 1967; Mermin and Wagner, 1966 ). It states that in 1D and 2D, continuous symmetries cannot be spontaneously broken at finite temperature in systems with interactions that are sufficiently short-ranged. The term continuous symmetry excludes for instance the Ising model but includes the isotropic Heisenberg and x–y models, as well as superconductivity, spin-density and charge-density waves. The lack of long-range order arises from the thermal population of the low-lying excitations. Consider first spin models for which the magnetization is a conserved quantity, i.e., ferromagnets. The T-dependence of the magnetization is given by:

(37) $M\left(T\right)-M\left(0\right)\sim -{\int }_0^{\infty }dE\frac{N\left(E\right)}{e^{E/k_{B}T}-1}\text{,}$

where N(E) is the density of states of the excitations. Assuming a dispersion E  ∼ k ⁿ in a system of D dimensions, one obtains N(E)   ∼ E ^(D−n)/n . The integral is then only convergent if (D  − n)/n  >   0. For ferromagnetic magnons we have n  =   2, so that there is long-range order in 3D, but not in 1D and 2D. For antiferromagnets the magnetization alternates from site to site and the total magnetization is not conserved. The integrand has then an additional factor a(k), which for a linear dispersion, E  ∼ k, is inversely proportional to k. Hence, an antiferromagnet also has long-range order in 3D, but not in 1D and 2D (Schreiber, 2008; Wagner and Schollwoeck, 2011).

If the system is anisotropic, i.e., has an easy axis, the dispersion has a different form, e.g., E  = A  + Ck ² with A and C being positive constants. The integral (37) now converges and the magnetic order is thus stabilized by anisotropy in all dimensions. Long-range interactions (e.g., dipolar interactions) usually give rise to less dispersive excitations (n is small) and the integral (37) converges. Hence, magnetic order may be stabilized by the long range of the interactions.

In summary, 2D models that are ordered in the ground state but for which the Mermin–Wagner–Hohenberg theorem predicts no long-range order at finite T, have a QCP. In this case the quantum fluctuations suppress the long-range order for T  >   0. In Section 5 we review how quantum fluctuations may lead to quantum criticality in systems with itinerant electrons, in particular heavy fermions.

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ON SYMPLECTIC REDUCTION IN CLASSICAL MECHANICS

J. Butterfield , in Philosophy of Physics, 2007

2.1.3 Noether's theorem

The core idea of Noether's theorem, in both the Lagrangian and Hamiltonian frameworks, is that to every continuous symmetry of the system there corresponds a conserved quantity (a first integral, a constant of the motion). The idea of a continuous symmetry is made precise along the following lines: a symmetry is a vector field on the state-space that (i) preserves the Lagrangian (respectively, Hamiltonian) and (ii) "respects" the structure of the state-space.

In the Hamiltonian framework, the heart of the proof is a "one-liner" based on the fact that the Poisson bracket is antisymmetric. Thus for any scalar functions f and H on a symplectic manifold (M, ω) (and so with a Poisson bracket given by eq. 18), we have that at any point x ∈ M

(19) $\begin{array}{ccc}{X}_f\left(H\right)\left(x\right)\equiv \left\{H,f\right\}\left(x\right)=0& \text{iff}& 0=\left\{f,H\right\}\left(x\right)\equiv X_H\left(f\right)\left(x\right)\end{array}\cdot$

In words: around x, H is constant under the flow of the vector field X_f (i.e. under what the evolution would be if f was the Hamiltonian) iff f is constant under the flow X_H. Thinking of H as the physical Hamiltonian, so that X_H represents the real time-evolution (sometimes called: the dynamical flow), this means: around x, X_f preserves the Hamiltonian iff f is constant under time-evolution, i.e. f is a conserved quantity (a constant of the motion).

But we need to be careful about clause (ii) above: the idea that a vector field "respects" the structure of the state-space. In the Hamiltonian framework, this is made precise as preserving the symplectic form. Thus we define a vector field X on a symplectic manifold (M, ω) to be symplectic (also known as: canonical) iff the Lie-derivative along X of the symplectic form vanishes, i.e. ${\mathcal{L}}_{\text{X}}\omega =0$ . (This definition is equivalent to X's generating (active) canonical transformations, and to its preserving the Poisson bracket. But I will not go into details about these equivalences: for they belong to the theory of canonical transformations, which, as mentioned, I will not need to develop.)

We also define a Hamilton system to be a triple (M, ω, H) where (M, ω)is a symplectic manifold and H: M → ℝ, i.e. M ∈ F (M). And then we define a (continuous) symmetry of a Hamiltonian system to be a vector field X on M that:

(i)

preserves the Hamiltonian function, ${\mathcal{L}}_{\text{X}}H=0$ ; and

(ii)

preserves the symplectic form, ${\mathcal{L}}_{\text{X}}\omega =0$ .

These definitions mean that to prove Noether's theorem from eq. 19, it will suffice to prove that a vector field X is symplectic iff it is locally of the form X_f. Such a vector field is called locally Hamiltonian. (And a vector field is called Hamiltonian if there is a global scalar f: M → ℝ such that X = X_f. ) In fact, two results from the theory of differential forms, the Poincaré Lemma and Cartan's magic formula, make it easy to prove this; (for a vector field on any symplectic manifold (M, ω), i.e. (M, ω) does not need to be a cotangent bundle).

Again writing d for the exterior derivative, we recall that a k-form α is called:

(i)

exact if there is a (k − 1)-form β such that α = dβ; (cf. the elementary definition of an exact differential);

(ii)

closed if dα = 0.

The Poincaré Lemma states that every closed form is locally exact. To be precise: for any open set U of M, we define the vector space Ω ^k (U) of k-form fields on U. Then the Poincaré Lemma states that if α ∈ Ω ^k (M) is closed, then at every x ∈ M there is a neighbourhood U such that α _U ∈ Ω ^k (U) is exact.

Cartan's magic formula is a useful formula (proved by straightforward calculation) relating the Lie derivative, contraction and the exterior derivative. It says that if X is a vector field and α a k-form on a manifold M, then the Lie derivative of α with respect to X (i.e. along the flow of X) is

(20) ${\mathcal{L}}_X\alpha ={\text{di}}_X\alpha +{\text{i}}_X\text{d}\alpha \cdot$

We now argue as follows. Since ω is closed, i.e. dω = 0, Cartan's magic formula, eq. 20, applied to ω becomes

(21) ${\mathcal{L}}_X\omega \equiv {\text{di}}_X\omega +{\text{i}}_X\text{d}\omega ={\text{di}}_X\omega \cdot$

So for X to be symplectic is for i _X ω to be closed. But by the Poincaré Lemma, if i _X ω closed, it is locally exact. That is: there locally exists a scalar function f: M → ℝ such that

(22) ${\text{i}}_X\omega =df\text{\hspace{0.17em}i}\cdot \text{e}\cdot X=X_f\cdot$

So for X to be symplectic is equivalent to X being locally Hamiltonian.

Thus we have

Noether's theorem for a Hamilton system If X is a symmetry of a Hamiltonian system (M, ω, H), then locally X = X_f; so by the anti-symmetry of the Poisson bracket, eq. 19, f is a constant of the motion. And conversely: if f: M → ℝ is a constant of the motion, then X_f is a symmetry.

We will see in Section 6.2 that most of this approach to Noether's theorem, in particular the "one-liner" appeal to the anti-symmetry of the Poisson bracket, eq. 19, carries over to the more general framework of Poisson manifolds. For the moment, we mention an example (which we will also return to).

For most Hamiltonian systems in euclidean space ℝ³, spatial translations and rotations are (continuous) symmetries. Let us consider in particular a system we will discuss in more detail in Section 2.3: N point-particles interacting by Newtonian gravity. The Hamiltonian is a sum of two terms, which are each individually invariant under translations and rotations:

(i)

a kinetic energy term K; though I will not go into details, it is in fact defined by the euclidean metric of ℝ³, and is thereby invariant; and

(ii)

a potential energy term V; it depends only on the particles' relative distances, and is thereby invariant.

The corresponding conserved quantities are the total linear and angular momentum. ⁶

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On the Principle of Least Action and Its Role in the Alternative Theory of Nonequilibrium Processes

Michael Lauster , in Variational and Extremum Principles in Macroscopic Systems, 2005

3 Callen's principle

To make GFD a workable concept, an important question has to be answered. If the compilation of the generics is the primary task of the construction of the theory then what—except from the observation of exchange mechanisms—could be used to find the extensive variables for the Gibbs Fundamental Equation?

Here Callen's symmetry principle, first formulated in 1974 [8], provides a powerful tool. In Callen's words, it reads

"The primary theorem, relating symmetry considerations to physical consequences, is Noether's theorem. According to this theorem every continuous symmetry transformation … implies the existence of a conserved function. … The most primitive class of symmetries is the class of continuous space-time transformation. The (presumed) invariance of physical laws under time translation implies energy conservation; spatial translation symmetry implies conservation of momentum. [p. 62]".

and further:

"The symmetry interpretation of thermodynamics immediately suggests, then, that energy, linear momentum and angular momentum should play fully analogous roles in thermodynamics. The equivalence of these roles is rarely evident in conventional treatments, which appear to grant the energy a misleadingly unique status …, it is evident, that in principle, the linear momentum does appear in the formalism in a form fully equivalent to the energy, … [p. 65]".

Obviously, conserved quantities play a distinguished role for the set of generics according to Callen's principle. Variables mapping conserved generics are the privileged members of the set of coordinates. Prominent examples from physics are the total energy or the linear and angular momentum obeying respective conservation laws.

However, in open systems even the values of conserved quantities may be altered by transfer processes over the boundaries of an object. Every conserved quantity possesses another extensive quantity as a partner providing these transfers. For example, the body forces are the source of changes for the linear momentum as well as the net electrical current is for the charge of the body. These partner coordinates also have to appear in the set of variables.

Furthermore, generics exist that show broken symmetries like the volume or the entropy. Volume and entropy are born members of the set of variables: the volume is needed to define the boundaries of the body over which the transfers take place. The entropy ensures that the amount of energy needed to form a coherent body from N-many particles is considered.

Thus, Callen's principle delivers conditions enabling the identification of significant co-ordinates for the GFR.

However, since neither GFD nor AT refer to conserved quantities as construction elements a principle will be required allowing conservation qualities to be introduced by means of mathematical procedures in order to avoid such characteristics as metaphysical add-ons of both the theories.

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SYMMETRIES AND INVARIANCES IN CLASSICAL PHYSICS

Katherine Brading , Elena Castellani , in Philosophy of Physics, 2007

8.4 Status of symmetries

Are symmetries ontological, epistemological, or methodological in status? It is clear that symmetries have an important heuristic function, as discussed above (Section 5) in the context of relativity. This indicates a methodological status, something that becomes further developed within the context of quantum theory. We can also ask whether we should attribute an ontological or epistemological status to symmetries.

According to an ontological viewpoint, symmetries are seen as "existing in nature", or characterizing the structure of the physical world. One reason for attributing symmetries to nature is the so-called geometrical interpretation of spatiotemporal symmetries, according to which the spatiotemporal symmetries of physical laws are interpreted as symmetries of spacetime itself, the "geometrical structure" of the physical world. Moreover, this way of seeing symmetries can be extended to non-external symmetries, by considering them as properties of other kinds of spaces, usually known as "internal spaces". ⁸¹ The question of exactly what a realist would be committed to on such a view of internal spaces remains open, and an interesting topic for discussion.

One approach to investigating the limits of an ontological stance with respect to symmetries would be to investigate their empirical or observational status: can the symmetries in question be directly observed? We first have to address what it means for a symmetry to be observable, and indeed whether all symmetries have the same observational status. Kosso [2000] arrives at the conclusion that there are important differences in the empirical status of the different kinds of symmetries. In particular, while global continuous symmetries can be directly observed — via such experiments as the Galilean ship experiment — a local continuous symmetry can have only indirect empirical evidence. ⁸²

The direct observational status of the familiar global spacetime symmetries leads us to an epistemological aspect of symmetries. According to Wigner, the spatiotemporal invariance principles play the role of a prerequisite for the very possibility of discovering the laws of nature: 'if the correlations between events changed from day to day, and would be different for different points of space, it would be impossible to discover them' [Wigner, 1967]. For Wigner, this conception of symmetry principles is essentially related to our ignorance (if we could directly know all the laws of nature, we would not need to use symmetry principles in our search for them). Such a view might be given a methodological interpretation, according to which such spatiotemporal regularities are presupposed in order for the enterprize of discovering the laws of physics to get off the ground. ⁸³ Others have arrived at a view according to which symmetry principles function as "transcendental principles" in the Kantian sense (see for instance [Mainzer, 1996]). It should be noted in this regard that Wigner's starting point, as quoted above, does not imply exact symmetries — all that is needed epistemologically (or methodologically) is that the global symmetries hold approximately, for suitable spatiotemporal regions, so that there is sufficient stability and regularity in the events for the laws of nature to be discovered.

As this discussion, and that of the preceding Subsections, indicate, the differences between various types of symmetry become important before we have ventured very far into interpretational issues. For this reason, much recent work on the interpretation of symmetry in physical theory has focussed not on general questions, such as those sketched above, but on addressing interpretational questions specific to particular symmetries. ⁸⁴

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Symmetry and Symplectic Reduction

J.-P. Ortega , T.S. Ratiu , in Encyclopedia of Mathematical Physics, 2006

The Symmetries of a System

The standard mathematical fashion to describe the symmetries of a dynamical system (see Dynamical Systems in Mathematical Physics: An Illustration from Water Waves) $X\in \mathfrak{X}\left(M\right)$ defined on a manifold $M\left(\mathfrak{X}\right(M)$ denotes the Lie algebra of smooth vector fields on M endowed with the Jacobi–Lie bracket $\left[\cdot ,\cdot \right]$ ) consists in studying its invariance properties with respect to a smooth Lie group $\text{Φ}:G\times M\to M$ (continuous symmetries) or Lie algebra $\text{ϕ}:\mathfrak{g}\to \mathfrak{X}\left(M\right)$ (infinitesimal symmetry) action. Recall that Φ is a (left) action if the map $g\in G\mapsto \text{Φ}\left(g,\cdot \right)\in \text{Diff}\left(M\right)$ is a group homomorphism, where $\text{Diff}\left(M\right)$ denotes the group of smooth diffeomorphisms of the manifold M. The map ϕ is a (left) Lie algebra action if the map $\xi \in \mathfrak{g}\mapsto \varphi \left(\xi \right)\in \mathfrak{X}\left(M\right)$ is a Lie algebra antihomomorphism and the map $(m,\xi )\in M\times \mathfrak{g}\mapsto \varphi \left(\xi \right)\left(m\right)\in TM$ is smooth. The vector field X is said to be G-symmetric whenever it is equivariant with respect to the G-action Φ, that is, $X\circ {\text{Φ}}_g=T{\text{Φ}}_g\circ \text{X}$ , for any $g\in G$ . The space of G-symmetric vector fields on M is denoted by $\mathfrak{X}{(M)}^G$ . The flow F _t of a G-symmetric vector field $X\in \mathfrak{X}(M{)}^G$ is G-equivariant, that is, $F_t\circ {\text{Φ}}_g={\text{Φ}}_g\circ F_t$ , for any $g\in G$ . The vector field X is said to be $\mathfrak{g}$ -symmetric if $[\varphi (\xi ),X]=0$ , for any $\xi \in \mathfrak{g}$ .

If $\mathfrak{g}$ is the Lie algebra of the Lie group G (see Lie Groups: General Theory) then the infinitesimal generators ${\xi }_M\in \mathfrak{X}(M)$ of a smooth G-group action defined by

$\begin{array}{cc}{\xi }_M(m):=\frac{\text{d}}{\text{d}t}{|}_{t=0}\text{Φ}(\text{exp}t\xi ,m),& \xi \in \mathfrak{g},m\in M\end{array}$

constitute a smooth Lie algebra $\mathfrak{g}$ -action and we denote in this case $\varphi (\xi )={\xi }_M$ .

If $m\in M$ , the closed Lie subgroup $G_m:=\left\{g\in G|\text{Φ}\left(g,m\right)=m\right\}$ is called the isotropy or symmetry subgroup of m. Similarly, the Lie subalgebra ${\mathfrak{g}}_m:=\{\xi \in \mathfrak{g}|\varphi (\xi )(m)=0\}$ is called the isotropy or symmetry subalgebra of m. If $\mathfrak{g}$ is the Lie algebra of G and the Lie algebra action is given by the infinitesimal generators, then ${\mathfrak{g}}_m$ is the Lie algebra of $G_m$ . The action is called free if $G_m=\left\{e\right\}$ for every $m\in M$ and locally free if ${\mathfrak{g}}_m=\{0\}$ for every $m\in M$ . We will write interchangeably $\text{Φ}\left(g,m\right)={\text{Φ}}_g\left(m\right)={\text{Φ}}^m\left(g\right)=g\cdot m$ , for $m\in M$ and $g\in G$ .

In this article we will focus mainly on continuous symmetries induced by proper Lie group actions. The action Φ is called proper whenever for any two convergent sequences ${\left\{m_n\right\}}_{n\in \mathbb{N}}$ and $\{g_n\cdot m_n:=\text{Φ}(g_n,m_n){\}}_{n\in \mathbb{N}}$ in M, there exists a convergent subsequence $\{g_{n_k}{\}}_{k\in \mathbb{N}}$ in G. Compact group actions are obviously proper.

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Symmetry Breaking in Field Theory

T.W.B. Kibble , in Encyclopedia of Mathematical Physics, 2006

Goldstone Bosons

The Goldstone theorem states that spontaneous breaking of any continuous global symmetry leads inevitably (except, as we discuss later, in the presence of long-range forces) to the appearance of massless modes – the Goldstone bosons.

The proof is straightforward. Associated with any continuous symmetry there is a Noether current satisfying the continuity equation ${\partial }_{\mu }{\hat{j}}^{\mu }=0$ and such that infinitesimal symmetry transformations are generated by the spatial integral of ${\hat{j}}^0$ . The fact that the symmetry is broken means that there is some scalar field $\hat{\varphi }\left(x\right)$ whose vacuum expectation value $〈0\left|\hat{\varphi }\left(0\right)\right|0〉$ is not invariant under the symmetry transformation. Hence,

[14] $\underset{v\to 0}{lim}\text{i}{\int }_v{d}^{3}x〈0\left|\left[{\hat{j}}^0\left(x\right),\hat{\varphi }\left(0\right)\right]\right|0〉{|}_{x^0=0}\ne 0$

Moreover, the time derivative of this integral is

[15] $\begin{array}{l}\underset{v\to 0}{lim}\text{i}{\int }_v{d}^{3}x〈0\left|\left[{\partial }_0{\hat{j}}^0\left(x\right),\hat{\varphi }\left(0\right)\right]\right|0〉{|}_{x^0=0}\ne 0\\ =-\underset{v\to 0}{\text{lim}}\text{i}{\int }_{\partial v}{\text{ds}}_k〈0\left|\left[{\hat{j}}^k\left(x\right),\hat{\varphi }\left(0\right)\right]\right|0〉{|}_{x^0=0}=0\end{array}$

where $\partial \mathcal{V}$ is the bounding surface of $\mathcal{V}$ . This vanishes because the surface integral is zero – in a relativistic theory, because the commutator vanishes at spacelike separation, and more generally in the absence of long-range interactions because it tends rapidly to zero at large spatial separation.

Now, inserting a complete set of momentum eigenstates $|n,\mathit{p}〉$ in [14], we can see that there must exist states such that $〈n,\mathit{p}\left|\hat{\varphi }\left(0\right)\right|0〉\ne 0$ , with p ⁰→0 in the limit | p |→0, that is, massless modes.

One can see this more directly in the U(1) model above. Consider a vacuum state |0〉 such that $〈0\left|\hat{\varphi }\right|0〉=\eta /\sqrt{2}$ is real. Then it is useful to shift the origin of ϕ by writing

[16] $\varphi \left(x\right)=\frac{1}{\sqrt{2}}[\eta +{\phi }_1(x)+\text{<mi mathvariant="bold" is="true">i</mi>}{\mathit{\phi }}_2(\mathit{x}\left)\right]$

where φ₁ and φ2 are real. Then the Lagrangian becomes

[17] $\begin{array}{l}\mathcal{L}=\frac{1}{2}[{\stackrel{.}{\phi }}_1^2-{\left(\nabla {\phi }_1\right)}^2+{\stackrel{.}{\phi }}_2^2-{\left(\nabla {\phi }_2\right)}^2-\text{λ}{\eta }^{\text{2}}{\phi }_1^2\\ -\text{λ}\eta {\phi }_{\text{1}}\left({\phi }_1^2+{\phi }_2^2\right)-\frac{1}{4}\text{λ}\left({\phi }_1^2+{\phi }_2^2\right)]\end{array}$

Evidently, the field φ, corresponding to radial oscillations in ϕ, is massive, with mass $\sqrt{\lambda }\eta$ . But there is no term in ${\phi }_2^2$ , so φ₂ is massless.

In the case of spontaneous symmetry breaking of nonabelian symmetries, there may be several Goldstone bosons, one for each broken component of the continuous symmetry. In our theory with symmetry group G=O(N), the possible values of the vacuum expectation value at T=0 are $〈0_{\mathbf{n}}\left|\hat{\varphi }\left(0\right)\right|0_{\mathbf{n}}〉\approx \eta \mathbf{n}$ , where n is an arbitrary unit vector. In this case, for given n , there is an unbroken symmetry subgroup

[18] $H=\left\{R\in O\left(N\right):R\mathit{n}=\mathit{n}\right\}=O\left(N-1\right)$

and the number of broken symmetries is

[19] $\text{dim}G-\text{dim}H=N-1$

Thus, the radial component of ϕ is massive, and there are N−1 Goldstone bosons, the N−1 transverse components.

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