Discretisation and Diffeomorphismsrelativity.phys.lsu.edu/ilqgs/bahr032310.pdf · Discretisation and Diﬀeomorphisms ... case the eom for tn is automatically satisﬁed. ⇒ S

Introduction Reparametrisation-invariant systems in 1D Improved and perfect actions Discrete Hamiltonian framework Triangulations Summary

Discretisation and Diffeomorphisms

Benjamin Bahr

Albert Einstein InstituteAm Muhlenberg 1

14476 Golm

Work (partially in progress) with Bianca Dittrich, Philipp Hoehn,

Song He, Ralf Banisch, Sebastian Steinhaus

1 / 31


Outline:

◮ Introduction◮ Gauge symmetries: Reparametrisation invariance in GR◮ Discretisations and (gauge) symmetries

◮ Reparametrisation-invariant systems in 1D◮ Breaking of rep-inv◮ Improved and perfect actions

◮ Gauge symmetries in GR discretised a la Regge◮ Diffeos vs. Vertex displacement symmetries◮ Improving the action: 3D with Λ 6= 0, 4D

◮ Discretised canonical formulation◮ Matches (breaking of) symmetries in discrete covariant setting◮ Constraints ⇔ ’pseudo-constraints’

◮ Triangulations and triangulation-independence

◮ Summary

2 / 31


Gauge symmetry in GR

Diff-invariance in GR as gauge symmetry

◮ ’12: realisation of Einstein that initialconditions do not determine solution tohis field equations uniquely (holeargument).

◮ Dirac: Different solutions with sameinitial data should be physically thesame.

◮ Solutions unique up to space-timediffeomorphisms!

times1 s2

Phase space (space of initial data)

◮ Principle of background-independence: No metric singled out (Diff (M) as gaugegroup).

◮ Canonical formulation: Constraint algebra (”hypersurface-deformation algebra”)

{D(X ),D(Y )} = D([X ,Y ]), {D(X ),H(N)} = H(LXN),

{H(N),H(M)} = D(q−1(NdM −MdN))

⇒ Diffeo-symmetry intimately related to dynamics of GR!

3 / 31



Discretisation

Before quantization, one usually discretises systems (easier to quantise, deal withproduct of ”fields at a point”)

Unfortunately, upon discretisation, Poisson algebra relations are usually changed.Note: not always bad! For example the E -fields in LQG:

◮ Continuum: {EI (x),EJ(y)} = 0

◮ Integrated over face f : {EI (f ),EJ(f )} = ǫIJKEK (f )

But when the fields are associated to (gauge) symmetries, change of Poisson structureusually viewed as anomaly

◮ In QFT: Lattice discretisation changes algebra of the Pµ ⇔ breaksPoincare-invariance

◮ In 3 + 1 GR: No anomaly-free version of discretised Hypersurface-deformationalgebra known (e.g. Loll gr-qc/9708025, Thiemann gr-qc/9705017, hep-th/0005232, Dittrich 0810.3594 [gr-qc])

◮ In 2 + 1 LQG with Λ 6= 0: Discretisation along plaquettes produes ”mild”anomaly (Perez, Pranzetti 1001.3292[gr-qc])

4 / 31



Discretising GR

For instance Regge gravity: GR on triangulation (starting point for Quantum Reggecalculus, Spin foam quantization)

◮ Replace smooth metric gµν bypiecewise linear-flat metric g∆(encoded in edge lengths le oftriangulation)

le

gµν

⇒

◮ Piecewise flat metrics approximate smooth ones, but have different orbit sizeunder diffeos.

⇒ Fate of diffeomorphism-symmetry in the discrete theory?

5 / 31



Aim of this talk

◮ Describe, in which way gauge symmetry is lost in discretised Gravity (Regge), andwhich problems this causes.

◮ Argue, that the following three problems are connected:

Find discrete actionwith exact gauge

symmetry

Find an anomaly-freediscrete canonicalconstraint algebra

Find a discrete actionwhich is independent

of triangulation

6 / 31


Parametrise 1D systems


Reparametrisation-invariant systems in 1D: Close to situation in GR

◮ Take usual system in 1D without gauge symmetries: L(q, q) = 12q2 − V (q),

det ∂2L∂qi∂qj

6= 0

◮ Parametrise: treat time t as variable and regard evolution w.r.t. auxilliaryparameter s: q(t) → q(s), t(s). Dynamics is governed by action:

L(q, t, q′, t′) :=

(

1

2

(q′)2

(t′)2− V (q)

)

t′

◮ The new system now is reparametrisation-invariant: q(s), t(s) solutions forcertain boundary data, then also q(s), t(s) for any s = s(s).

⇒ Boundary data q(si ), t(si ), q(sf ), t(sf ) does not determine unique solution ofequations of motion (only up to reparametrisation). System has gauge symmetryin that sense. ⇔ Analogue of diff-symmetry in GR.

◮ Generically: Eom for t(s) is satisfied automatically. ⇒ Choose arbitrary t(s),then q(s) is uniquely determined.

7 / 31



Discretise parametrised 1D systems

◮ Discretise the variables: q(s), t(s) → tn, qn, n = 0, . . . ,N + 1. Discretise action:

Sdiscr =N∑

n=0

(

1

2

(qn+1 − qn)2

(tn+1 − tn)2− V

(

qn + qn+1

2

))

(tn+1 − tn)

(Equivalent to replace smooth solutions by piecewise linear ones).

◮ Equations of motion:

∂Sdiscr

∂tn=

∂Sdiscr

∂qn

!= 0 unique solution!

⇒ gauge-invariance broken! Unlike continuum t(s) (which is arbitrary), tn arefixed!

◮ Therefore the Hessian H = ∂2Sdiscr∂xn∂xm

has no zero Eigenvalue (where x = {t, q}).

However: In limit N → ∞, N out of the 2N eigenvalues of H approach zero. Incontinuum limit, gauge symmetry is restored in that sense.

8 / 31



Exception:

◮ There is an exception, for when also the discretised action Sdiscr exhibits gaugesymmetry: Whenever the piecewise linear dynamics coincides with the continuumdynamics, i.e. for V = 0 (⇔ linear dynamics q = const):

tn

qn

Figure: V = 0: tn arbitrary, qnuniquely fixed by boundary dataand tn

tn

qn

Figure: V 6= 0: tn and qn uniquelyfixed by boundary data.Nobodycan read this...

◮ If V = 0, then the discrete dynamics coincides with the continuum one. In thiscase the eom for tn is automatically satisfied. ⇒ Sdiscr has exact gaugesymmetries from the continuum (tn arbitrary, qn fixed by boundary data and tn).

9 / 31



The perfect action

◮ The breaking of gauge symmetry in discrete theory can be traced back to thechoice of the discrete action Sdiscr, which effectively replaces smooth dynamicsby piecewise linear ones. In fact, one can write down a discrete action whichexhibits exact gauge symmetry (perfect action): (see e.g. Marsden, West Act.Num.10 (2001))

Sperf =N∑

n=0

∫ sn+1

sn

ds L(

q(s), t(s), q′(s), t′(s))

where q(s), t(s) are solutions of continuum dynamics with boundary data tn, qn,tn+1, qn+1.

◮ The Hessian∂2Sperf

∂xn∂xmhas N zero eigenvectors (exhibits exact gauge symmetries of

continuum: half of the variables are gauge). Reproduces exact continuumdynamics.

10 / 31



Improving the action via coarse graining

◮ The perfect action can be constructed by coarse graining process (see e.g. BB,

Dittrich 0907.4323 [gr-qc]): Sdisc(q0, t0, . . . , qN , tN ) → Simp(q0, t0, . . . , qN , tN )

Simp =N∑

n=0

S(n)imp

(qn, tn, qn+1, tn+1)tn, qn

tn, qn

⇓

solve

insert

where

S(n)imp

(qn, tn, qn+1, tn+1) = Sdiscr(q0, t0, . . . , qM , tM )∣∣

∣

∂Sdiscr∂qn

=∂Sdiscr

∂ tn=0, qn=q0,...,tn+1=tM

is the value of the discrete action Sdiscr on a solution of a further refinement of theinterval [tn, tn+1] into M pieces.

◮ The improved action is defined on the coarse lattice, but incorporates dynamics

of fine lattice. ⇒∂2Simp

∂xn∂xmhas N small Eigenvectors.

◮ Perfect action arises as limit of infinite refinement:

limM→∞

Simp = Sperf

11 / 31



Problems with discretisation:

◮ For naively discretised reparametrization-invariant theories, both tn and qn arefixed by dynamics, but in the continuum limit only qn(tn) is predicted by theory⇒ Mismatch of degrees of freedom (e.g. correlation function between tn andtn+1 becomes pure gauge in continuum).

◮ Too many variables integrated over in path integral (divergencies)!

◮ Different size of gauge orbits causes problems for perturbation theory Dittrich, Hoehn,

0912.1817 [gr-qc]

◮ Numerical issue: very small Eigenvalues of Hessian H (for large N) causenumerical problems.

◮ In lattice field theories: discretisation causes lattice artifacts which often rangeover several lattice sites. Bietenholz hep-lat/9709117

In lattice field theories, these issues are well-known, and dealt with by improving theaction (replacing Sdiscr with another action Simp, which captures gauge degrees offreedom better).

◮ For lattice field theories, improved (and perfect) actions are highly sought after.Symanzik Nucl.Phys.B226:187,1983 Gupta hep-lat/9807028, Bietenholz, Wiese hep-lat/9510026 They areconstructed using renormalization group techniques.

◮ Improving the action reduces artifacts significantly.

◮ Perfect actions always exist for asymptotically free theories.

12 / 31


Vertex displacements and breaking of gauge symmetries in Regge gravity

Recap: Regge gravity

◮ In Regge calculus Regge ’61, one replaces smooth metric by piecewise-linear flatmetrics living on a triangulation τ , which consists of D-dim. simplices σ.

Discretised variables: edge lengths le .Fh: volume of D − 2 subsimplex h

(”hinge”)θσh: interior dihedral angle at h in σ

ǫh := 2π −∑

σ⊃h θσh

deficit angleψh := π −

∑

σ⊃h θσh.

h

τle

∂τ

θσh

The discretised action is given by

SRegge =∑

h∈τ◦

Fhǫh − Λ∑

σ

Vσ +∑

h∈∂τ

Fhψh

13 / 31



Gauge symmetry for Regge in 3D

◮ The equations of motion for 3D Regge gravity (for Λ = 0) are ǫe = 0, so thesolutions are locally flat metrics - coincides exactly with solutions of continuumdynamics! As a result, the discrete dynamics exhibits exact gauge symmetries(compare to the V = 0 case for 1D systems)

◮ Symmetry corresponds todisplacement of vertices, position ofwhich can be freely chosen (3gauge directions per vertex).Symmetry can be derived fromdiscrete Bianchi identities, and iseven an off-shell symmetry. Freidel,

Louapre gr-qc/0212001

◮ Same kind of symmetry exists for 4D Regge and Λ = 0, whenever boundaryconditions are chosen which lead to flat solutions ǫ∆ = 0 in the interior. ⇒ again,matches continuum solution Rµ

νσρ = 0, again vertex displacement symmetry ispresent (4 gauge directions per vertex). Can (in linearised theory) be shown tocorrespond to action of 4-diffeomorphisms in continuum. Rocek, Williams Phys.Lett.B 104

(1981)

14 / 31



Breaking of gauge symmetries in Regge

In the continuum limit of 4D Regge the vertex displacement symmetry is restored.However, whenever discrete solutions do not reproduce continuum dynamics exactly,one has breaking of gauge symmetries. Examples:

◮ 3D Regge with Λ 6= 0. The equations ofmotion are

ǫe − Λ∑

σ⊃e

∂Vσ

∂le= 0

⇒ Unique solution for fixed boundary data:No gauge symmetries.

◮ 4D Regge with Λ = 0: BB, Dittrich: 0905.1670 [gr-qc]

Consider triangulation resulting from tentmoves A → A′ (one inner vertex). Theboundary conditions determine the deficitangle ǫ∆ at interior triangle ∆. Whenever theangle ǫ∆ is non-zero (solution withcurvature), then the minimal eigenvalue λmin

of the Hessian does not vanish ⇒ no gaugesymmetries! In fact: λmin ∼ ǫ2∆.

��

��

��

x y

A

A′

∆

-0.02 -0.01 0.01 0.02

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

ǫ∆

λmin

15 / 31



Improving the action for 3D Regge with Λ 6= 0

BB, Dittrich: 0907.4323 [gr-qc]

◮ For this case the coarse graining procedurecan be applied to improve the Regge action.In the limit of infinite refinement, theimproved action converges to the Reggeaction for simplices of constant curvatureκ = Λ.

⇒

SRegge =∑

e

leǫe − Λ∑

σ

Vσ ⇒ S(κ)Regge

=∑

e

leǫ(κ)e + 2κ

∑

σ

V(κ)σ

◮ The perfect action in fact S(κ)Regge

possesses exact gauge symmetries (vertex

displacement symmetry: 3 gauge dof per vertex).

◮ Equations of motion are ǫ(κ)e = 0: Solutions are metric of constant curvature κ.

Regge action with flat simplices can be seen as linear approximation:

S(κ)Regge

= SRegge + O(κ2).

16 / 31



Improving the action for 4D Regge:

◮ In general: (improved and) perfect actions are non-local, but (in case of Regge)hopefully triangulation-independent. BB, Dittrich, He, work in progress

◮ In 4D, one can only hope to compute perfect action approximately. PerturbRegge around flat solution (regular 4D hypercubes) and improve order by order:

S = ǫ21

2See′ l

(1)e l

(1)e′

+ ǫ3(

See′ l(1)e l

(2)e′

+1

3!See′e′′ l

(1)e l

(1)e′

l(1)e′′

)

+ . . .

↓ improve

S = ǫ21

2Simp

ee′l(1)e l

(1)e′

+ ǫ3(

Simp

ee′l(1)e l

(2)e′

+1

3!See′e′′ l

(1)e l

(1)e′

l(1)e′′

)

+ . . .

↓ improve

S = ǫ21

2Simp

ee′l(1)e l

(1)e′

+ ǫ3(

Simp

ee′l(1)e l

(2)e′

+1

3!Simp

ee′e′′l(1)e l

(1)e′

l(1)e′′

)

+ . . .

17 / 31



Consistency conditions for perturbation

le = l(0)e + ǫl

(1)e + ǫ2l

(2)e + . . .

S = ǫ21

2See′ l

(1)e l

(1)e′

+ ǫ3(

See′ l(1)e l

(2)e′

+1

3!See′e′′ l

(1)e l

(1)e′

l(1)e′′

)

+ . . .

◮ Because l(0)e describes flat solutions, the Hessian See′ possesses zero vectors y

ge

(gauge symmetries corresponding to vertex translations Rocek, Williams Phys.Lett.B 104

(1981)). To linear order:

See′yg

e′= 0

l(n)e = l

(n)p yp

e + l(n)g yg

e (Separation into physical and gauge modes)

⇒ l(1)p determined, l

(0)g , l

(1)g free

◮ To higher order, however:

⇒ l(2)p determined, l

(1)g , l

(2)g free

plus: condition on background gauge variables l(0)g !

18 / 31



Consistency conditions for perturbation

◮ Since higher order terms break gauge symmetry, solving for higher orderdetermines background gauge! ⇒ Due to fact that the solution we perturbaround has higher symmetry than generically. Perturbative solutions s (withcurvature) have smaller gauge orbit than flat solution (which has 4 gauge dof pervertex).

G = gauge orbit of flat solution. v1and v2 gauge equivalent, but perturba-tion around v1 is consistent, around v2 isnot!

G s

v2v1

◮ One can show (Dittrich, Hoehn 0912.1817 [gr-qc]): The eom fixing background gaugeparameters are equivalent to

(

∂S

∂leyge

)

∣

∣

∣2nd order

= yge

(

∂SHJ

∂l(0)e

)

∣

∣

∣2nd order

19 / 31


Discrete Hamiltonian framework

The same issue in canonical language:

In canonical formulation, gauge symmetries of the action arise as constraints.Situation in Regge?

◮ In BB, Dittrich 0905.1670 [gr-qc] a Hamiltonianframework was presented, which matchesexactly presence (or breaking) ofgauge-symmetries of action. Based on tentmoves (Sorkin et al gr-qc/9411008).

lne

ln+1e

tn

◮ Tent moves leave combinatorics of 3D hypersurface invariant. Regge action forpart between n-th and n + 1-st hypersurface: Sn. Generating function forcanonical transformation lne → ln+1

e , pne → pn+1e

pnt := −∂Sn

∂tnpne := −

∂Sn

∂lne

pn+1t := −

∂Sn

∂tn+1pn+1e := −

∂Sn

∂ln+1e

◮ Equations of motion: pnt = pn+1t = 0, pne = −pn+1

e .

20 / 31



Dynamics and constraints:

Dittrich, Hoehn 0912.1817 [gr-qc]

◮ For four-valent vertex being evolved by tent move: Only flat solution ǫ∆ = 0!

Ce = pne −∑

∆⊃e

∂a∆

∂lneψ∆(lne ) = 0

Equation only between variables lne , pne at time step n. ⇒ constraint

◮ Higher-valent vertex being evolved by tent move: More than flat solutions.

Ce = pne −∑

∆⊃e

∂a∆

∂lneψ∆(lne , l

n+1e ) = 0

Equation involves variables lne , ln+1e at different time steps. Momenta depend

(weakly) on ln+1e ⇒ ”pseudo-constraints”

21 / 31



”Dynamics” for one simplex

Dittrich, Ryan 0807.2806 [gr-qc], Dittrich, Hoehn 0912.1817 [gr-qc]

◮ Example: Triangulation consistingof one four-simplex.

◮ Canonical transformation:

A∆ = A∆(le)

p∆ =∂le

∂A∆pe

◮ Constraints Ce → C∆:

C∆ = p∆ + ψ∆

◮ ⇒ Constraints fix momenta to be exterior angles (as computed by the lengths).

◮ Are fist class!

◮ Generate deformations of (boundary) hypersurface.

◮ Ten lengths (+ momenta), ten constraints: ⇒ no dof left! (flat interior)

22 / 31



Linearised constraints for 4D linearised Regge

Dittrich, Hoehn 0912.1817 [gr-qc]

◮ Linearised 4D Regge around flat sector: le = l(0)e + ǫl

(1)e + . . .,

pe = p(0)e + ǫp

(1)e + . . .. Since Hessian has null vectors

∂2SRegge

∂le∂le′yg

e′= 0, one

also has (linearised) constraints:

Cg = pn,(1)e yg

e + yg

e′

∂

∂le′

∑

∆⊃e

∂a∆

∂leψ∆

∣

∣

∣le=l(0)e

ln,(1)e

◮ Constraints give relation between intrinsic and extrinsic geometry

◮ Are first class! (despite very complicated form of dihedral angles)

◮ Generate linearized deformation of hypersurface (via vertex translations):Hamiltonian and diffeomorphism constraints

◮ Are preserved by linearized tent move dynamics

23 / 31


Triangulations and triangulation-independence

Independence of action on triangulation?

Since the perfect action should exhibit exact gauge symmetries, it should reproducethe continuum dynamics. Therefore, it should, if computed w.r.t. differenttriangulations with same boundary, yield the same result on solutions (be”triangulation-independent”).

◮ Classification of four-manifolds (Pfeiffer gr-qc/0404088): Path integral for GR should beDiff-invariant of four-manifolds. Theorem by Pachner:

Smooth 4-manifolds / diffeos ⇔ piecewise linear 4-manifolds / Pachner moves

◮ Here Triangulation is not seen as approximation or cut-off, but to contain wholeinformation of smooth manifold!

◮ Triangulation-independence not because of ”topological theory”, but ratherbecause of action on renormalization-group fixed point (”quantum perfectaction”).

24 / 31


Triangulations and triangulation-independence

Triangulation-(in-)dependence in Regge

◮ In 3D Regge with Λ = 0. X

◮ In 3D Regge with Λ 6= 0 if one uses

perfect action S(κ)Regge

. X

◮ In 4D, whenever there is some partof the interior which is flat (includesall 1− 5 moves of simplices). X

◮ In 4D, the Regge action is not

invariant under the 3− 3 move.The difference of SRegge

before/after 3− 3 move is againquadratic in ǫ∆ for (either) interiortriangle ∆!

⇒

τ1 τ2

-0.2 -0.1 0.1 0.2

0.1

0.2

0.3

0.4

0.5

ǫ∆

Sτ1 − Sτ2

⇒ Triangulation-independence in Regge is realised to the same extent in which actionis invariant under vertex translations.

25 / 31


Summary + Outlook

Summary

◮ Diffeomorphism-symmetry crucial for General Relativity ⇒ fate upondiscretisation important for quantisation!

◮ Generically, upon discretisation, reparametrization-invariance (⇔Diffeomorphism-symmetry) is lost, whenever discrete dynamics exactly reproducescontinuum dynamics.

◮ In particular, gauge symmetries are broken in Regge calculus, except for 3D withΛ = 0 and 4D flat sector (many different piecewise linear flat metric approximatethe same (diffeomorphic) smooth metric).

◮ Either corresponds to anomaly ⇒ cure! Or: many more microscopic dof thanmacroscopic dof, and diff-symmetry is only realized in continuum limit.

26 / 31


Summary + Outlook

Improved and perfect actions

◮ Presence (or absence) of gauge symmetry is result of action (and of way in whichit is discretised). Restore symmetries by ”improving” action (coarse-grainingprocedure: incorporate dynamics on finer triangulation into coarser triangulation).

Sdiscrquantize−→

∫

Dg∆ e iSdiscr

coarse grain ↓ ↓ renormalize

Sperfquantize−→

∫

D[gµν ] eiSperf

◮ Investigation of diff-invariant discrete actions worthwhile: Conditions for pathintegral? (compare LOST-thm in LQG)

◮ Complementary analysis to ”sum over triangulations” (GFT approach!):Effectively include sum over (some) finer triangulations in Simp.

27 / 31


Summary + Outlook

◮ Improving action restores gauge symmetries:◮ 1D discrete rep-invariant mechanics: it works!◮ 3D Regge with Λ 6= 0: it works!◮ 3D Regge with scalar field? Dittrich, Banisch, work in progress

◮ 4D Regge with Λ 6= 0: it works (for constantly curved sector)!

◮ For Regge, there is a discrete canonical analysis which exactly matches discretecovariant formulation, using tent moves:

gauge symmetries present ⇔ exact 1st-class constraints

gauge symmetries broken ⇔ ’pseudo constraints’

(crf ”consistent discretisation” approach Gambini-Pullin ’03-’05)

◮ Furthermore: Regge is triangulation (in-)dependent to the same extent in whichit is invariant under vertex translations. (3− 3 move: ∆S ∼ ǫ2)

28 / 31


Summary + Outlook

⇒ Hints that the following three problems are in fact related:

Find discrete actionwith exact gauge

symmetry

Find an anomaly-freediscrete canonicalconstraint algebra

Find a discrete actionwhich is independent

of triangulation

29 / 31


Summary + Outlook

Outlook

◮ Improve linearised 4D Regge: BB, Dittrich, He: work in progress Subtle: higher orders fixbackground gauge parameters!

◮ For general quadratic actions of variables xi :

Sdiscr =1

2xiMijxj

Gauge directions: Mijygi= 0. Coarse grain variables: XI = bIixi

Simp =1

2XIMIJXJ

with

MIJ = bIiypi(M−1)pqy

qjbJj , where Mpq := y i

pMijyjq

◮ MIJbIiygi= 0 ⇒ Gauge directions are preserved under coarse graining.

◮ Restoration of triangulation independence in perfect limit?

◮ Locality of perfect action?

30 / 31


Summary + Outlook

Outlook:

◮ In quantum theory: coarse graining (renormalization group flow) should lead topath integral which is exact ”projector” onto physical Hilbert space.

◮ ⇒ Works so far for 1D discretised quantum mechanics for (an-)harmonicoscillator (perturbatively and non-perturbatively) BB, Dittrich, Steinhaus, work in progress

◮ Quantise perfect action S(κ)Regge

for 3D Regge with Λ 6= 0: Relation to

Turaev-Viro?

◮ Turaev-Viro as renormalization group fixed point of ”Ponzano-Regge +ΛV ”?

31 / 31

Documents

Discretisation and Diffeomorphismsrelativity.phys.lsu.edu/ilqgs/bahr032310.pdf · Discretisation and Diﬀeomorphisms ... case the eom for tn is automatically satisﬁed. ⇒ S