Symmetry breaking, massless excitations & massive “gauge” fields

one-forms may not commute at different points on the loop. In physics, (2.98) is called the Wilson loop and is an important gauge invariant quantity in (thermal) gauge theories. The spacetime derivative is supplemented by a gauge connection in order to define gauge invariant derivatives and the corresponding field strength tensor is then interpreted as the curvature form D Φ“d Φ´igAΦ“d Φ´igrA,Φs, igF “D^D“ rD,Ds, (2.99) which yields the dynamical terms for the gauge field. The conservation law for currentscannot be determined by Noether’s theorem since it only holds for global symmetries. Instead, con-served currents arise from the identityd²“0which is also valid for non-Abelian gauge groups if we take the proper covariant extensioni.e. the corresponding Bianchi identities. Note that the corresponding charges are not conserved since for non-Abelian gauge groups the gauge field are not neutral/uncharged. The full gauge invariantUp1qLagrangian of quantum electrodynamics describing charged particles/electrons (q“ ´e) and photons can then be written as

L_QED“ ´1

4F²`Ψpiγ^µDµ´mqΨ

“ ´pBµA_ν´ BνA_µqpB^µA^ν´ B^νA^µq `iΨγ^µBµΨ´eΨγ^µA_µΨ (2.100) where the gauge covariant derivative induces an interactionqΨγ^µA_µΨbetween the fermionic fields (electrons) mediated by the gauge connection/field (photons). In quantum theories of ele-mentary particles there exist two more forces, with no classical counterpart, the weak interaction transmitted by anSUp2qgauge field and the strong interaction described by quantum chromody-namics, anSUp3qgauge theory. Equation (2.95) for general non-AbelianSUpNqgauge groups reads

L_YM“ ´1

2trF²“ ´1

4F^aµνF_µν^a where F_µν^a “ BµA^a_ν´ BνA^a_µ`gYMf^abcA^b_µA^c_ν (2.101) and the corresponding field theories are called Yang-Mills theories. In particular the gauge field excitations or gauge bosons, being in the adjoint representation, carry charges for non-Abelian gauge theories. The adjoint representation of theUp1qgauge group is trivial and hence the pho-ton, and all other possible excitations transforming in the adjoint representation, are uncharged gXg^´1ÝÑ e^iqλXe^´iqλ “X ñ rX, Ys “0. (2.102) yet

AÝÑ e^iqλAe^´iqλ `e^iqλ d e^´iqλ “A´iqdλ . (2.103) A nice book discussing Yang-Mills theories and its application to particle physics is [55]. Classi-cally, we can also consider gravity as a gauge theory (in the broader sense that gauge symmetries describe mathematical redundancies) but here we need to be careful. The active diffeomorphism invariance of general relativity is not a true redundancy but rather a change of reference frame.

Observers in different reference frames do measure different results for the same event. But events are independent of the underlying parametrization of the spacetime manifold as shown by Einstein’s hole argument; see for example [46,56].

• Explicit symmetry breaking:

The symmetry is explicitly broken in the Lagrangian of the theory,i.e. a symmetry breaking term is added, which will give rise to gapped low-energy excitations. For critical phenome-na/phase transitions this approach cannot connect different phases by varying a (thermo-dynamic) parameter since the symmetry is always broken.

• Spontaneous symmetry breaking:

In this case only the symmetry of the vacuum state is broken whereas the Lagrangian remains invariant. In quantum theories a symmetry of the vacuum must also be a symmetry of the Hamiltonian butnot vice versa. An invariant Hamiltonian which commutes with conserved charges allows for two possible realizations

“

H

^,

Q

a

‰“0ñ

$’

’&

’’

%

Q

a|0y “0ô e^iεaQa |0y “ |0y

@0ˇˇΦ^:

H

Φˇˇ0D₂₁

“A

0ˇˇˇΦ˜^:

H

Φ^˜ˇˇˇ0E

Q

a|0y ‰0 e.g. xδ_aΦy “ix0| r

Q

a,Φs |0y ‰0 . (2.104) The first case can be generalized to show the invariance of arbitrary correlation functions leading to the Ward-Takahashi identities hinted at in the InfoboxNoether’s Theorem on page40. Since for allε^athe vacuum state is invariant, all generators must annihilate the vacuum state. Here the symmetric vacuum state possess a finite energy gap as well. In the second case at least one of the correlation functions is not invariant under the symmetry, usually the vacuum expectation value of a field, which is identified as the order parameter, by acquiring a fixed value. Additionally, a non-zero two point correlation function of the order parameter imply long range order

|x´xlim¹|Ñ8

@ΦpxqΦpx¹qD

‰0 (2.105)

and the spontaneously broken symmetry phase describes a ordered system on a global scale.

Strictly speaking, a spontaneous symmetry breaking can only occur in the thermodynamic limit N Ñ 8, V Ñ 8 while ^N{^V “ const. Mathematically, the order parameter is a logarithmic derivative of the partition function which is composed of a finite sum of weighting factors and thus cannot be a singular function. Only in the continuous limit a singularity may emerge.

Physically, thermal and/or quantum fluctuations will destroy the uniform (fixed) value of the order parameter and thus dynamically restore the symmetry (see also Section2.3.4and Infobox Coleman-Mermin-Wagner Theoremon page53). Thermal fluctuations vanish in the thermody-namic limit and for quantum field theories the infinite number of degrees of freedom decouple different vacuum/ground states since the mixed matrix elements of the Hamiltonian are zero and there is no quantum superposition of different vacuum states anymore. So the vacuum state characterized by the order parameter is stable and it takes an infinite time to fluctuate into a different vacuum state.

The fact that the vacuum expectation is not invariant under the symmetry leads to the celebrated Goldstone theorem for spontaneously broken continuous global symmetries:²²

21This follows directly from A

0ˇˇˇΦ^:HΦˇˇˇ0E

“A

0ˇˇˇe^´iεQΦ˜^:e^iεQHe^´iεQ Φ e˜ ^iεQ ˇˇˇ0E

“A

0ˇˇˇΦ˜^:HΦ^˜ˇˇˇ0E yielding vacuum/ground states with degenerate energy.

22While a discrete symmetry may also give rise to degenerate (non-contingent) vacuum states, it is not possible to move from one symmetry-breaking vacuum state to another with zero energy fluctuations. For an infinite system (e.g. in

Physical system Broken symmetry Excitation Quasi-particle

Fluids Galilean symmetry density/sound waves phonons

Solids translational & rotational lattice vibrations crystal phonons

Magnets rotational spin waves magnons

Superfluids globalUp1q density waves phonons

QCD chiral-flavor quark-antiquark pions^˚

Table 2.4. The Nambu-Goldstone excitations are universal features of physical systems exhibit-ing spontaneous symmetry breakexhibit-ing. The massless/gapless modes are indepen-dent of the microscopic details as long as the (microscopic) interactions are suf-ficiently short-ranged. The corresponding quasi-particles are characterized by the same quantum numbers as the respective generators. Note that the chiral symme-try breaking yields massive pseudo-Goldstone bosons due to the quark condensate’s explicit symmetry breaking.

Goldstone’s Theorem

A spontaneously broken continuous symmetry in a system with sufficiently short-ranged in-teractions leads to massless excitations along the directions spanned by the generators of the symmetry (more precisely, the coset space) thatdo notleave the vacuum/ground states invariant because

BV `

e^´iεaQaΦ˘ Bε^a

ˇˇ ˇˇ ˇε^a“0

“ BV BΦ^j

B`

e^´iεaQa˘ BΦ^j

ˇˇ ˇˇ ˇε^a“0

“ ´iBV

BΦ^j

Q

aΦ^j“^! 0

ñ B BΦ^k

ˆBV BΦ^j

Q

aΦ^j

˙

“ B²V

BΦ^kBΦ^j

Q

aΦ^j`B

Q

aΦ^j BΦ^k

BV BΦ^j “0

ñ B²V BΦ^kBΦ^j

ˇˇ ˇˇ

Φ^j“Φ^k“Φ^p0q

Q

aΦ^p0q“

M

_jk²

Q

aΦ^p0q“0

where the mass operator squared is defined as the curvature of the potential functionalV and the vacuum states are defined as the minima ofV,i.e.^BV{^BΦj|Φ^j“Φ^p0q “0. For all generators

Q

aΦ^p0q‰0the corresponding eigenvalue of

M

_jk² are zero. The remaining generators, leaving the broken symmetry vacuum state invariant, correspond to the residual symmetry gauge group. Excitations along these directions possess an energy gap related to the shape of the potential (i.e. the non-zero eigenvalues of

M

_jk²). The manifold of degenerate vacuum states may be parametrized by a continuous field describing the effective low-energy dynamics of the massless excitations.

The massless low-energy excitations are called Nambu-Goldstone modes and can be related to quasi-particles in various systems, c.f. Table 2.4. In Fourier representation these modes will manifest themselves as zero momentum poles of the corresponding correlation functions. For

the thermodynamic limit) the energy cost is infinite and the system is said to be rigid with respect to this broken symmetry.

continuous phase transitions the onset of the ordered phase can be probed by computing the pole structure of the respective susceptibilities in the unbroken phase. Upon approaching the critical point the zero momentum pole of the susceptibilities related to the order parameter fluctuations will cross the origin of the complex frequency plane indicating an instability due to the expansion around the “false vacuum”. Therefore, assuming well behaved low-energy effective theories, the critical point can be determined without computing thermodynamic properties such as the free energy in the broken and unbroken phase. This method will come into its own when applied to holographic systems, where calculations in the ordered phase may become intractable, seee.g.

Section4.5and5.3.

In principle gauge symmetries/redundancy cannot be broken spontaneously. This is the essence of Elitzur’s theorem [57] and as a consequence we cannot construct a Landau type local order parameter. The different phases in a gauge theory can be characterized by a gauge invariant physical quantity, the Wilson loops introduced in (2.98). The high-temperature phase displays an exponential decay determined by the enclosed area of the loop

WrCs „ e^{´const.¨ArearCs}ˇˇˇ

TąTc

(area law) (2.106)

whereas the low-temperature phase is related to the length of the Wilson loop WrCs „ e^{´const.¨|C|}

ˇˇ ˇTăTc

(perimeter law) (2.107)

For a (lattice)Up1qgauge theory the gapped/confined phase with linear flux tubes connecting opposite charges is described by the area law whereas the deconfined/Coulomb phase exhibit the perimeter law. In non-perturbative/strongly coupled QCD the Wilson loops can be used to distinguish the hadronic phase from the deconfined quark-gluon plasma phase,c.f. Figure1.2.

The area law of the Wilson loop for the hadronic phase can be understood as being proportional to the flux tube area which in turn is proportional to the separation of antiquark/quark pairs.

Mesons as bound states of antiquark/quark pairs contain two Wilson loops almost canceling each other which yields a small area enclosed by the total loop. The Wilson loop in the deconfined phase is proportional to the path traced out by “free” antiquark/quark pairs which is character-ized by the perimeter law.

Nonetheless, we can choose a particular gauge where the gauge symmetry may be hidden in the vacuum states, such that the gauge transformations of the gauge group become trivial. This symmetry breaking mechanism follows from the observation that we can “gauge away” the “arti-ficial” or spurious Nambu-Goldstone mode by a redefinition of the gauge field. This is the famous Anderson-Higgs mechanism explaining massive photons in superconductors and the mass of the weak force gauge bosons.

Anderson-Higgs Mechanism

The Anderson-Higgs mechanism gives rise to massive “gauge” fields for a particular choice of gauge that encodes only the true physical degrees of freedom. The “symmetry breaking”

turns the original gauge field into a massive gauge invariant vector field by removing the unphysical massless Goldstone mode.

The number of degrees of freedom is conserved and the action is still manifestly gauge in-variant. Thus, the term “spontaneously broken gauge symmetry” should be read as “gauge symmetry/transformation becomes trivial”. Historically, the terms Anderson-Higgs mecha-nism and spontaneously broken gauge symmetry are used synonymously due to their formal equivalence. For an elucidating discussion on these matters see the excellent review about spontaneously violated gauge symmetries in superconductors [58].

Experimentally, there is strong evidence that our universe is actually in a condensed phase where a non-zero vacuum expectation value of a scalar field provides the electroweak symmetry break-ing mechanism in the standard model and thus gives rise to massive gauge bosons and generates masses of the elementary particles. Last year a new bosonic particle was discovered at the Large Hadron Collider in Geneva which seems to have the properties predicted by the standard model calculations. The mass of the particle is about126^GeV{^c2but if it is a scalar spin zero particle has yet to be ascertained.²³ If the new particle is the so-called Higgs particle, the collective excita-tion of the condensed scalar field, the particle/field content of the standard model of elementary particle physics would be complete. Studying the new particle more closely could yield more information about the properties of the (current) phase or state of our universe. In principle it could be possible that the Higgs field is of composite nature and the corresponding collective excitations are an emergent phenomenon.²⁴ Funnily enough, in condensed matter physics a Higgs-like mode has been detected (even prior to the discovery at CERN) in an quantum anti-ferromagnet displaying a dimerized disorder to magnetic long range order transition [61] and recently in an ultracold atoms system close to the critical point of the superfluid-insulator tran-sition [62]. A Higgs mode can only exist in a relativistic theory but in condensed matter the low-energy excitations are non relativistic (i.e. the mass gap is not equal to the dynamic mass of the excitation). However, in quantum phase transitions the ordered and disorder phase as well as the quantum critical point is described by relativistic Lagrangians. In addition to the gapless Goldstone modes, quantum critical systems possess gapped low-energy excitations with a mass gap proportional to the curvature of the minimized potential. In the superfluid-insulator exper-iment [62] the experexper-imental visibility is obscured due to the low-dimensionality of the system.

The Higgs mode can decay into multiple Goldstone modes and the respective amplitude of such a process in two dimensions has an infrared singularity.²⁵

Im Dokument Gauge/Gravity duality (Seite 64-68)