Symmetries in quantum field theories

As explained in the previous section, the continuous phase transition between ordered and dis-ordered (symmetric) phases is described by spontaneous symmetry breaking. Let us recapitulate symmetries in physics, first in classical field theory and finally in quantum field theory. If the action is invariant under a certain set of symmetry operations belonging to a symmetry group, we say the corresponding theory is invariant under this symmetry transformations

Φⁱpxq ÝÑΦⁱpxq ´iε^agaΦⁱpxq “Φⁱpxq `iε^aFⁱ

apxq, ñδaSrΦs “iε^a

ż

d^dx δSrΦs δΦⁱpxqFⁱ

apxq“^! 0,

(2.87)

whereg^a denotes the generators of the symmetry group andFⁱ

a the action of the group gener-ators on the fields. Note that the Hamiltonian can be viewed as the generator of the dynamical evolution of the system since the equations of motion always leave the action invariant (Hamil-ton’s principle of stationary action)

δSrΦs “

ż

d^dx δLpΦ,BµΦq “

ż

d^dx

„BL BΦⁱ ´ Bµ

ˆ BL BpBµΦⁱq

˙

δΦⁱ, (2.88)

for vanishing variations on the boundaryΦⁱ|B“0.

We can also distinguish several types of symmetries:

• Global symmetries, acting on the entire system at every point, such as the Poincaré sym-metry or the internal isospinSUp2qsymmetry.

• Local symmetries, depending on local coordinate representations of the base manifold such as the spacetime,i.e.ε“εpxq.

Additionally, we can discriminate between

• Physical symmetries, act on physical states and commute with the microscopic Hamilto-nian. More precisely, the physical states transform in some representation of the symmetry group (e.g. for the Lorentz symmetry this classifies all one particle states). The Hamiltonian commutes with all generators and thus the eigenvalues are called quantum numbers that label the physical states.

• “Gauge” symmetries¹⁹characterizing the mathematical redundancy in describing the same physical state of the system. Thus, this symmetry relates redundant degrees of freedom in the theory. Observables and physical states cannot transform under gauge symmetries.

Of course there are local and global gauge redundancies/symmetries, but physical symmetries can only be global, because physical local symmetries can only exist for free theories (e.g. in effective theories with Gaussian fixed point or in theT Ñ 8 limit of a statistical system). All symmetries can be continuous, described by Lie groups, or discrete, described by a symmetry group with finite order. Futhermore, there are purely internal symmetries acting only on internal degrees of freedom. In Table2.2and2.3we list some physical systems and their corresponding symmetries. For global continuous symmetries there is a powerful theorem relating conserved

“charges” (constants of motion) and “currents” to the respective generators of the symmetry group:

Noether’s Theorem

Under an arbitrary symmetry operation

Φⁱpxq `iε^apxqFⁱ

apxq, the variation of the action is given by

δ_aSrΦs “ ż

d^dx J_a^µpxqBε^apxq Bx^µ

Every continuous global symmetry yields a classical conserved current (not related to forces/interactions) for each symmetry transformation labeled bya

19A better parlance for gauge symmetry is gauge redundancy since gauge symmetry is no symmetry in a strict mathe-matical sense and cannot be broken in principle.

Discrete symmetries n Classical system Quantum system

GlobalZ_n

1 Potts model —

2 Ising model quantum Ising model

8 XY model quantum rotor

boson Hubbard model

Local gaugeZ_n 2 Ising lattice

(topological)quantum liquids gauge theory

Table 2.2. Overview of some common discrete symmetries encountered in physical theories/

models. The different globalZ_n models are sometimes calledn-state Potts model, vector Potts model or more generally the clock model. The Ising and the XY model may be realized as continuous vector symmetries as well,c.f. Table2.3.

δaSrΦs “ ´ ż

d^dx ε^aBµJ_a^µpxq“^! 0

for variations (2.88) vanishing at the boundary. There exists a corresponding conserved charge

Q_a “ ż

d^DxJ_a⁰ where BtQ_a“ ´ ż

d^DxBiJ_aⁱ “ ´ż ` Bd^Dx˘

iJ_aⁱ«0

for sufficiently decaying currents near the boundary. Note that Noether’s theorem relies on the on-shell action and the classical equations of motion. The off-shell quantum version of Noether’s theorem implicitly assumes expectation values and therefore the symmetry applies only to correlation functions. The related operator identities using equations of motion for correlation functions are called Ward-Takahashi identities.

If the Lagrangian or even the Lagrangian density is invariant we can write explicit formulae for the conserved currents²⁰

J_a^µ“ ˆ BL

BpBµΦⁱqBνΦⁱ´δ^µ_νL

˙

δx^ν_a´ BL BpBµΦⁱqF ⁱ

a “ BL

BpBµΦⁱqigaΦⁱ´Lδx^µ_a, (2.89) where the action of the generators on the fields is generalizes to

igaΦⁱpxq “ BµΦpxqⁱδx^µ_a´F ⁱ

a . (2.90)

For an exhaustive discussion of symmetries and conserved currents see [54]. ApplyingNoether’s Theorem on page 40to spacetime translation we find the definition of the conserved energy-momentum tensor

T^µ_ν “ L

BpBµΦⁱqBνΦⁱ´δ^µ_νL, (2.91)

20Only the second term follows from a invariant Lagrangian density, since spacetime variationsx^µÝÑx`ε^aδax^µdo not leave the Lagrangian density invariant.

Continuous symmetry n Classical system Quantum system

GlobalOpnq

1 Ising model quantum Ising model

2 XY model Op2qquantum rotor model

boson Hubbard model 3 Heisenberg model Op3qquantum rotor model

4 Higgs sector toy model —

GlobalUpnq 1 vector/axial symmetry —

2 — isospin symmetry

GaugeUpnq

1 electrodynamics quantum electrodynamics

2 — weak interactions

3 — quantum chromodynamic

Table 2.3. Continuation of the overview of common symmetry groups in physical mod-els. Enhancing the discrete symmetries to continuous symmetries allows for true vector models. A very prominent model with continuous symmetry is the standard model of elementary particle physics given by the product group SUp3q ˆSUp2q ˆUp1q{Z₆. There are also approximate symmetries such as the SUp3qflavor and the chiral symmetry of QCDUp2qL ˆUp2qR “ SUp2qLˆSUp2qRˆ Up1qvectorˆUp1qaxial which are only realized for massless quarks. TheSUp2qL ˆ SUp2q^R is spontaneously broken by a finite quark condensate and due to its ap-proximate nature the corresponding Nambu-Goldstone bosons (the three pions) are massive,c.f. Table2.4.

where the corresponding charge is the conserved total energy and the spatial momentum because the time-spatial componentsT⁰ⁱ are the momentum density current and theT⁰⁰ component is the energy density. The complete discussion carries over to quantum field theories (replacing Poisson brackets with commutators) but with the caveat of so-called “anomalies”. These anoma-lies change the functional integral measure which in turn changes the action or the Lagrangian of the quantum field theory. The name “anomalies” is more or less a misnomer since their origin lies in the fact that we need to take the Jacobian of a coordinate transformation into account. As stated above, symmetries in classical physics leave the action invariant (2.87) but symmetries in quantum physics need to be realized on the level of the quantum states and hence the partition function needs to be invariant, so the functional integral measure must be preserved in addition to the classical action

δSrΦs “0 ø δZ“δ ż

DΦe^´SrΦs “0. (2.92) Therefore, anomalies are symmetries which hold at the classical level but are broken at the quantum level because

DΦÝÑJ DΦ˜‰DΦ . (2.93)

A well known anomaly is the axial anomaly. Classically the current corresponding to the global axial symmetry

ΨÝÑ e^iεγ5Ψ in L “iΨγ^µBµΨ where Ψ“` ψL, ψR

˘T

, (2.94)

is conserved, but in quantum field theory, there are non-zero contributions arising from what are known as triangular diagrams due to quantum fluctuations [50]. We will see in Section2.3.3 that quantum fluctuations can not only destroy symmetries but also dynamically restore them.

Theories with gauge redundancy are described by the action of the gauge group generators In particular we need to modify the derivatives by introducing a gauge connection relating differ-ent “gauges” of the physical quantities to each other. These gauge fields naturally give rise to interactions transmitted between the fields transforming under the gauge group. A famous (and to my knowledge the only classical example) is Maxwell’s theory of electromagnetic interactions with aUp1qgauge group. The theory can be written entirely in the language of differential forms allowing immediate extension to curved spacetime if necessary

AÝÑA`dλpxq ñ L_em“ ´1

4pdAq²ÝÑ ´1

4pdA`d²λpxqq²“L_em, (2.95) Turning to quantum physics, we show that the phase invariance (describing a true gauge re-dundancy) of a complex quantum field implies the existence of an interaction mediated by a gauge field. Charged matter can be described (microscopically) by quantum operators and their respective excitations from the vacuum state such as charged fermionic particles (electrons) transforming in the fundamental representation of theUp1q. In a heuristic picture, we can view the different states of an electron connected by theUp1qsymmetry

ΨÝÑ e^´ieλpxqΨ, length of Õ “

e^´ieλpxq

“1, position of Õ “λpxq, (2.96) as internal indicators (displaying the phase of the wavefunction) carried by each electron. Now the photon exchanged by two electrons plays the role of transmitting the position of the indica-tor from an electron to the other in such a way that both electrons can retain their respective choice of phase. Therefore for an external observer it is impossible to extract the actual phase of an electron, because any measurement done by interchanging photons can only detect phase changes. This “reality” of the gauge field was demonstrated by the experimental realization of the Aharonov-Bohm effect. It can be shown that local phases accumulated by contractible loops are represented by forces (here the magnetic force) but global phases of non-contractible loops are not related to any classical force [4]

ϕlocal“iq

¿

dx¨A“iq ż

dn¨B, ϕglobal“iqnwΦmagnetic flux, (2.97) wherenwdenotes the winding number and the magnetic flux is confined to a small non-accessible region (e.g. a single flux tube), hence the loop is non-contractible. Nonetheless the free quantum particle will be subject to some interaction since its quantum properties are changed, but the clas-sical equations of motion are only affected by local phases generated by non-zero gauge fields.

Mathematically, the local phase is related to the holonomy of the gauge connection whereas the global gauge are related to the monodromy. In differential geometry parlance this holonomy can be viewed as a parallel transport around a closed loop

WrCs “tr

» –Pexp

˜ i

¿

C

A

¸fi

fl, (2.98)

where the trace is the character of the irreducible representations of the gauge group and P denotes the path ordering analogous to the time ordering in the path integral since the gauge

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].

Im Dokument Gauge/Gravity duality (Seite 59-64)