We motivate the concept of emergent gauge symmetry and discuss ways that this
concept can be tested. The key idea is that if a symmetry is emergent, one
should look for small violations of this symmetry because the underlying
fundamental theory does not contain the symmetry. We describe our recent work
implementing this idea in the gravity sector. We also describe the reasons why
violations of gauge symmetry may well be linked to violations of Lorentz
invariance.Comment: 5 pages, Invited talk presented at the Fifth Meeting on CPT and
  Lorentz Symmetry, Bloomington, Indiana, June 28-July 2, 201

Anber, Mohamed

Aydemir, Ufuk

Donoghue, John F.

English

arXiv

This is the pre-published version which is collected from arXiv.We motivate the concept of emergent gauge symmetry and discuss ways that this concept can be tested. The key idea is that if a symmetry is emergent, one should look for small violations of this symmetry because the underlying fundamental theory does not contain the symmetry. We describe our recent work implementing this idea in the gravity sector. We also describe the reasons why violations of gauge symmetry may well be linked to violations of Lorentz invariance

Donoghue, John

Anber, Mohamed M.

ScholarWorks@UMass Amherst

GAUGE NON-INVARIANCE AS TESTS OF EMERGENT GAUGE SYMMETRY

We motivate the concept of emergent gauge symmetry and discuss ways that this concept can be tested. The key idea is that if a symmetry is emergent, one should look for small violations of this symmetry because the underlying fundamental theory does not contain the symmetry. We describe our recent work implementing this idea in the gravity sector. We also describe the reasons why violations of gauge symmetry may well be linked to violations of Lorentz invariance

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics2010GAUGE NON-INVARIANCE AS TESTS OFEMERGENT GAUGE SYMMETRYJohn DonoghueUniversity of Massachusetts Amherst, donoghue@physics.umass.eduMohamed M. AnberUniversity of Massachusetts Amherst, manber@physics.umass.eduUfuk AydemirUniversity of Massachusetts Amherst, uaydemir@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationDonoghue, John; Anber, Mohamed M.; and Aydemir, Ufuk, "GAUGE NON-INVARIANCE AS TESTS OF EMERGENT GAUGESYMMETRY" (2010). Physics Department Faculty Publication Series. 242.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/242arXiv:1007.5049v1  [hep-ph]  28 Jul 2010July 29, 2010 0:6 WSPC - Proceedings Trim Size: 9in x 6in emergenceCPT101GAUGE NON-INVARIANCE AS TESTS OF EMERGENTGAUGE SYMMETRYJOHN F. DONOGHUEa∗, MOHAMED M. ANBERb and UFUK AYDEMIRcDepartment of Physics, University of Massachusetts,Amherst, MA 01003, USAa(∗speaker) E-mail: donoghue@physics.umass.edub Email: manber@physics.umass.educ Email: uaydemir@physics.umass.eduWe motivate the concept of emergent gauge symmetry and discuss ways thatthis concept can be tested. The key idea is that if a symmetry is emergent,one should look for small violations of this symmetry because the underlyingfundamental theory does not contain the symmetry. We describe our recentwork implementing this idea in the gravity sector. We also describe the reasonswhy violations of gauge symmetry may well be linked to violations of Lorentzinvariance.1. Emergence and emergent symmetryThe word “emergence” is appearing with increasing frequency in the par-ticle physics literature. While it can mean different things to different peo-ple, at its heart it implies that properties and degrees of freedom that oneobserves differ drastically from the actual underlying physics. A classic ex-ample is the emergence of phonons in materials. The underlying physics isatoms bumping into each other, but the low energy description is that ofwaves that satisfy the massless wave equation. These waves can be quan-tized and used to explain the heat capacity of materials, for example, but atthe smallest distance scales there in no such thing as a fundamental phononfield.There is a familiar example of emergence in particle physics - that ofpions and the chiral lagrangian. Here the underlying degrees of freedom arequarks and gluons with the interactions of QCD, yet the lightest fields arepions with a vastly different lagrangian,LQCD = −14FµνFµν + ψ¯i /Dψ → Leff =F 2pi4Tr[∂µU†∂µU ] (1)July 29, 2010 0:6 WSPC - Proceedings Trim Size: 9in x 6in emergenceCPT102with U = exp (iτ · pi/Fpi). If two of the quarks had been massless, thepions would also be massless and they would be active in atomic physics,well below the QCD scale. It would have taken an extraordinary leap ofimagination for the atomic physicists of the 1930’s to infer the underlyingstructure of QCD when confronted with these massless bosons.However, our subject here is actually somewhat different - that of “emer-gent symmetry”. The pion example above is not emergent symmetry. Thechiral symmetry of the pion lagrangian is a reflection of the underlyingchiral symmetry of QCD. However, symmetries can also emerge. As an ex-ample, consider the classic textbook derivation of the wave equation frommasses interacting with their neighbors along a one-dimensional array. Ifone takes the general interaction potential V (yi − yi−1) near its minimumit is approximately harmonic ∼ 12k(yi− yi−1)2. Taking the continuum limitleads to the 2D wave equation for waves on a string, i.e. the massless Klein-Gordon equation.S =∫dt∑i[12my˙2i − V (yi − yi−1)] →∫d2x12∂µφ∂µφ (2)where here ∂µ ≡ (∂t/vs, ∂x) with vs being the speed of sound. This latterform has symmetries that the original lagrangian did not have. The wavelagrangian is translation invariant, while the original one was not - this is aconsequence of the continuum limit. There is also an emergent Lorentz-likesymmetry - the wave lagrangian is invariant under Lorentz transformations,with vs in place of c of course. There is also a shift symmetry φ → φ + c,which keeps the wave massless, which is not really emergent. This is areflection of an underlying symmetry of the original system of shifting allthe yi by a constant, even though the wave field is not defined in the originallagrangian.This example can also be used to illustrate the key principle of thephenomenology of emergent symmetry. Since the symmetry is not exact,one should look for evidence of the violation of the symmetry. In the “waveson a string” case, we can see this explicitly. If we look at the next term inthe expansion of the potential about the minimum,V (yi − yi−1) =12k(yi − yi−1)2 + λ(yi − yi−1)4 + .... (3)when we take the continuum limit, the differences turn into spatial deriva-tives and we arrive at the lagrangianS =∫d2x[12∂µφ∂µφ+ λ¯(∂φ∂x)4](4)July 29, 2010 0:6 WSPC - Proceedings Trim Size: 9in x 6in emergenceCPT103where the last term involves only spatial derivatives. This then breaks theemergent Lorentz-like symmetry.This should be a general feature of emergent symmetry. If the low energysymmetry is not shared by the full theory, there will be some effects whichdo not have the symmetry. These symmetry violating terms are required tobe suppressed if the symmetry has indeed emerged, but their presence is aindicator.2. Thinking about emergent symmetryThe question then arises whether the symmetries of the Standard Modelcould be emergent. This of course would be a vastly different possibilitythan envisioned in the standard unification paradigm, where the low energysymmetries are part of larger symmetries at high energy.There is no complete model for the emergence of the Standard Modelsymmetries. However, there has been a modest body of work - more thancan be summarized accurately in the allotted write-up for this talk, so thatspace allows only the briefest of references.1Even without a complete model for emergent symmetries, there areaspects of the theory that we understand from other work.On the positive side is the theorem by Deser2 and others that says thatif one has a massless spin-two field which couples to energy and momentum,including its own energy-momentum, the result of iterating the couplingswill be general relativity. This is potentially an encouragement for emer-gence, as it implies that the full non-linear nature of general relativity couldbe the consequence of a seemingly simpler requirement of coupling the fieldto energy and momentum.On the potentially negative side for emergence is the Weinberg-Wittentheorem,3 which states that gravitons and Yang-Mills gauge bosons cannotbe emergent from an underlying theory with Lorentz invariance. This fol-lows from the requirement a Lorentz invariant emergent theory would leadto a Lorentz covariant energy-momentum tensor and gauge current. How-ever the energy-momentum tensor of the physical gravitons and the gaugecurrent of physical YM gauge bosons are not Lorentz covariant3 - althoughthis feature is hidden by the ability to make a gauge transformation on thefields simultaneously with a Lorentz boost. The Weinberg-Witten theoremthen appears to indicate that Lorentz invariance must also be emergent atthe same time as gauge symmetry.Another potential obstacle is the Nielsen-Ninomiya theorem4 that for-bids chiral fermions on a lattice. One might hope to avoid the Weinberg-July 29, 2010 0:6 WSPC - Proceedings Trim Size: 9in x 6in emergenceCPT104Witten theorem by considering a discretetized spacetime, which wouldget around the Lorentz invariance problem. However, in this case, chiralfermions are problematic.Also part of the emergence-Lorentz connection is the need for a universallimiting velocity (the speed of light) for all fields. In our experience withemergent fields, each carries its own velocity - phonons and magnons donot propagate at the same speed. However, to match our world all types offields need to have the same limiting velocity - a stringent requirement.3. PhenomenologyIf a symmetry is emergent, it makes sense to look for potential small vio-lations of that symmetry. Although much effort has gone into the study ofthe violation of discrete and global symmetries, very little work has beendone on the violation of gauge symmetries.Our published work in this area concerns gravity, in which case we areinterested in studying the violation of general coordinate invariance.5 Grav-itational physics has the potential to be more sensitive to emergence thanwould usual gauge invariance tests. This is because we expect that the signalof emergence would be suppressed and gravity is itself suppressed by twoinverse powers of the Planck mass. It is possible that a small signal could berelatively more visible in comparison to this already suppressed interaction.In addition, gravitational physics opens up vast time and distance scales,which could potentially be more revealing.In our approach, we considered possible modifications to the Lagrain-gian which contain two derivatives of the metric. This amounts to 5possible terms expressed using the connection, of which one of which isL3 = − gαγgβρgµνΓµαβΓνγρ. Because these are not generally covariant, theseare only consistent in a particular set of coordinates, which then forms aconstraint on allowable solutions. We studied the full set of Lagrangiansat linear order, and there are constraints on the coefficients of the variousLagrangian terms at the 10−3 level from the bending of light. There arealso constraints on the sign of some combinations of coefficients from theperturbative stability of the graviton propagator.We also took the Lagrangian L3 of the previous paragraph and per-formed the full matching to the PPN formalism.6 Our work cannot be donein the standard PPN gauge of that formalism, so we transformed the PPNgauge to the coordinates appropriate for our Lagrangian. The strongestconstraint comes from the orbital polarization of binary pulsars and givesa constraint on the coefficient of L3 at the 10−20 level, when normalized inJuly 29, 2010 0:6 WSPC - Proceedings Trim Size: 9in x 6in emergenceCPT105the same way as the Einstein-Hilbert action. This is quite strong, althoughwithout a fundamental underlying emergent theory the implication for thepossibility of emergence is not clear.We also are exploring the possibility that gauge symmetry violationmay come along with the violation of Lorentz invariance.7 This connectionis motivated by the Weinberg-Witten theorem - in emergent theories theenergy scale of Lorentz violation and the scale of emergence could be thesame. This connection may provide more sensitivity, because in many casesthe tests of Lorentz invariance are quite strong and can be extended togauge violating interactions.There are many possible directions that one can consider in the studyof emergent symmetry. While the unification paradigm has been pursuedfor several decades, the topic of emergent symmetry has been only lightlyexplored. That is part of the value of the subject - it provides a novelpathway for fundamental physics and one which may have new surprises.The possibility of testing the hypothesis exists, using probes which measurethe violation of gauge symmetry and/or general covariance.AcknowledgmentsThis work has been supported in part by the NSF grants PHY- 055304 andPHY - 0855119, and in part by the Foundational Questions Institute.References1. H.B. Nielsen, London Roy.Soc. 1983:0051.J. Ambjorn, R. Janik, W. Westra and S. Zohren, Phys. Lett. B 641, 94(2006) [arXiv:gr-qc/0607013].M. Levin and X.G. Wen, Rev. Mod. Phys. 77, 871 (2005).S. Weinfurtner, M. Visser, P. Jain and C.W. Gardiner, PoS QG-PH, 044(2007) [arXiv:0804.1346 [gr-qc]].G.E. Volovik, Phil. Trans. Roy. Soc. Lond. A 366, 2935 (2008)[arXiv:0801.0724 [gr-qc]].2. S. Deser, Gen. Rel. Grav. 1, 9 (1970) [arXiv:gr-qc/0411023].3. S. Weinberg and E. Witten, Phys. Lett. B 96, 59 (1980).4. H.B. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 20 (1981)5. M.M. Anber, U. Aydemir and J.F. Donoghue, Phys. Rev. D 81, 084059(2010) [arXiv:0911.4123 [gr-qc]].6. C.M. Will, “Theory and experiment in gravitational physics,” Cambridge,UK: Univ. Pr. (1993) 380 p7. M.M. Anber, U. Aydemir and J.F. Donoghue, (to appear)

JOHN F. DONOGHUE

MOHAMED M. ANBER

UFUK AYDEMIR

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GAUGE NONINVARIANCE AS TESTS OF EMERGENT GAUGE SYMMETRY

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Gauge non-invariance as tests of emergent gauge symmetry

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