Spin foam models and the classical action principle

For a large class of theories, the approach gives results in good agreement with known state sum models.

“As we shall see, for a large class of theories our approach gives results that are in a good agreement with the known state sum models.”

In a typical quantum state, the electric field is distributional and supported on the edges of a graph.

“Thus, in a typical quantum state, the electric field E is distributional with support on edges ei of the corresponding graph.”

When the action is evaluated on the distributional fields, it becomes a sum of integrals over lower-dimensional hypersurfaces.

“The action of the theory calculated on such distributional fields becomes a sum of certain integrals over the hypersurfaces of lower dimension.”

The method reproduces exactly the known spin foam models for BF theory and two-dimensional Yang-Mills theory.

“We find that it exactly reproduces the known models for BF and 2D Yang-Mills theories.”

• Also stated in: § y(h ei (A),---,h en (A)), · plain

  “We find that our approach exactly reproduces the state sum models of 2D Yang-Mills and BF theory in any dimension.”

For BF theory with a cosmological term in three and four dimensions, the first-order terms in the cosmological constant of the transition amplitude agree with those of known state sum models.

“For the BF theory with cosmological term in 3 and 4 dimensions we calculate the terms of the transition amplitude that are of the first order in the cosmological constant, and find an agreement with the corresponding first order terms of the known state sum models.”

• Also stated in: § y(h ei (A),---,h en (A)), · plain

  “For BF theories with cosmological term we compare the amplitude obtained within our approach with the one given by the corresponding state sum model by decomposing them into the power series in the cosmological constant and comparing the first order terms.”

The generating functional is independent of the chosen orientation of the wedges.

“However, the generating functional is independent of the orientation chosen.”

The integral of this distributional E over a wedge equals X_w for that wedge.

“Then the integral of such distributional E over a wedge of the triangulation is equal simply to X w corresponding to that wedge.”

The paper exactly computes the generating functional Z in the context of spin foam models.

“One of our main results is the exact computation, in the context of spin foam models, of the generating functional Z.”

For BF theories with a cosmological term, comparing first-order terms in the cosmological constant gives agreement with the corresponding state sum model.

“Again, we find an agreement.”

For topological theories, the result for a fixed collection of surfaces does not depend on that collection.

“In the case of topological theories, the result for a particular fixed collection of surfaces does not depend on the collection.”

The integral over X_w in (12) yields a function of exp Z_w and exp J_w proportional to the group delta-function peaked at exp J_w.

“In fact, it is not hard to see that this function is proportional to the 5-function of exp Z w peaked at exp J w .”

The generating functional is generally not invariant under refinement of the triangulation unless the theory is topological (unless J=0).

“The generating functional is generally not invariant under a refinement of the triangulation A unless the theory is a topological field theory (unless J = 0).”

Assuming M has a single connected component, the two-dimensional generating functional is given by (25).

“To describe the result of this integration we will assume that M has a single connected component.Then the generating functional in two dimensions is given by the following expression Z 2 (J, A) = ^(dim^nmm^mM © (25 )”

• AssumptionThe authors state this as an assumption: The manifold M has a single connected component for the two-dimensional generating functional result.

Tr(X_w Z_w) is gauge invariant only when exp Z is the holonomy around the wedge with base point at the d-simplex center.

“With this definition Tr(X w Z w ) is gauge invariant only when exp Z is defined as the holonomy around the wedge w whose starting and final point is the center C of the d-simplex, as in (6).”

The approach exactly reproduces the state sum models of two-dimensional Yang-Mills theory and BF theory in any dimension.

“We find that our approach exactly reproduces the state sum models of 2D Yang-Mills and BF theory in any dimension.”

Solving the E equations of motion from (34) and substituting back reduces the action to the standard Yang-Mills action (35).

“Solving equations of motion for E that follow from this action, and substituting the solution into the action, one can check that the action reduces to the standard Yang-Mills action SYM = -^I M d 2 xJgg ac g bd Tr(F ab F cd ), (35 )”

In the path integral, the Lie algebra variables X_w associated with the same dual face are constrained to have equal norm.

“Recall now that the integrals over X w are not independent: all X w that belong to one and the same dual face should have the same norm.”

The path integral (16) is equivalent to (12).

“Having discussed how one can calculate the path integral ( 16), let us now show that this path integral is, in fact, equivalent to (12).”

In two spacetime dimensions, Yang-Mills action depends only on a measure (area two-form) defined by the metric.

“However, in two spacetime dimensions, Yang-Mills action turns out to depend only on a measure (area two-form) defined by the metric on M.”

That coefficient must equal P(J_w), where P relates Lebesgue measure on the Lie algebra to Haar measure on the group.

“As we explain below, this function must be set to be equal to P(J W )^ where P is the function that relates Lebesgue measure on the Lie algebra and Haar measure on the group (see the Appendix B).”

Using [23], one can only argue that (16) equals P(J_w) delta(e^{Z_w} h e^{J_w} h^{-1}) for some group element h.

“Thus, strictly speaking, using the results of [23] one can only argue that ( 16) is equal to P(J w )8{e z ™he J ™h-1 ), (18 )”

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The restriction of E to a spatial slice is non-zero only along edges of the graph formed by the dual complex intersection.

“The restriction of E constructed as described above to a generic”

Z is obtained by integrating fields A and E that live on a fixed triangulation of spacetime.

“It is obtained by integrating over fields .A, E that”

The edges of Gamma_t are in one-to-one correspondence with the wedges.

“As we have just seen the edges of Tt are in one-to-one correspondence with the wedges.”

The constructed distributional E field has constant length along each edge.

“Note also that our construction of the distributional E field gives us E whose length is constant along each edge.”

After integrating over the h group elements, the generating functional takes the form (22).

“Integrating over all these edges, and making a simple change of variables to eliminate trivial integrations, we find z(j,A)=n/ <*&* ( 22 )”

For a wedge w with plane u=0, the distributional E field is du delta(u) X_w.

“Then the field E that has the correct distributional character is given by the expression du5(u)X w . (27 )”

The proportionality coefficient in that delta-function result can be a gauge-invariant function of J_w.

“The proportionality coefficient can be a gauge invariant function of J w .”

The path integral (16) is the partition function of BF theory on the disk with a distributional source J = delta(p) J_w.

“The path integral ( 16) is just a partition function of BF theory on the disk with the distributional source given by (17).”

The function in (12) can be written as P(J_w) times a sum over irreducible representations of dim j times the character chi_j(exp Z_w exp J_w).

“Thus, the function of Zw, Jw given by ( 12) can be written as p (Jw) YldimjXj(exp Z w exp J w ), (14 ) j”

The discretized action is a sum over wedges of Tr(F X_w) plus Tr(J X_w).

“Using (2),(3), we find that this discretized action is given by”

The generating functional is invariant under gauge transformations Z(gJg^{-1},A)=Z(J,A).

“The generating functional (11) satisfies: Z(gJg-1 ,A) = Z(J,A). (23 )”

For BF theory, introducing independent wedge variables does not change the theory when dual faces have disc topology.

“For the case of BF theory one can directly prove that the introduction of the independent wedge variables does not change the theory in the case the topology of dual faces a is that of a disc, which is what we will assume here.”

• AssumptionThe authors state this as an assumption: The dual faces have the topology of a disc.

In the group integration, only terms with equal spins for all wedges of the same dual face survive.

“Note that in the integration we just performed survived only the terms for which the spins of all wedges corresponding to one and the same dual face are equal.”

The expression (22) has a larger invariance parameterized by d+1 group elements per simplex, which contains the gauge transformation as a special case.

“However the expression ( 22) for the generating functional has a bigger invariance.Namely, let us denote by e+(w),e-(w) the two edges of the wedge w which meet at the center of the d-simplex h(w).”

• Also stated in: § Jw · plain

  “This transformation, which is parameterized by d + 1 group elements per simplex, contains the gauge transformation as its particular case.”

In the limit of small cosmological constant, the Turaev-Viro amplitude differs from the BF amplitude by a power of h.

“Thus, the Turaev-Viro amplitude, in the limit of small cosmological constant, differs from the BF amplitude by a power of”

One equation of motion from (38) states that A is the spin connection compatible with the triad E.

“One of the equations of motion that follows from (38) states that A is the spin connection compatible with the triad E.”

Each edge contributes j_e(j_e+1)/6 to the volume, where j_e is the spin labelling the edge.

“each edge contributes je{je + l)/6, where j e is the spin that labels the edge e.”

Not only tetrahedra but also edges and vertices of the triangulation contribute to the volume.

“It is interesting to note that not only tetrahedra t of A contribute to the volume, but also the edges e and vertices v.”

The volume (64) receives contributions from vertices, edges, and tetrahedra.

“The volume (64) has three types of contributions: (i) from vertices; (ii) from edges; (iii) from tetrahedra.”

Formally the path integral (45) equals the integral over A of delta(F).

“Indeed, formally the path integral (45) is equal to ' VA8(F). /=(48)”

The volume (64) is purely imaginary.

“Note that the volume turns out to be purely imaginary.”

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Each vertex contributes exactly 1/12 to the volume.

“One finds that each vertex contributes exactly 1/12”

In the discretized Turaev-Viro model, this power of h is replaced by h^V.

“In the discretized version of the theory, given by the Turaev-Viro model, this power of h is replaced by fi? v .”

The difference in integration measures between the two constructions is a power of h.

“The difference in the integration measures is a power of h.”

The first h-order term of the quantum (15j)-symbol is given by equation (78).

“one can easily check that the first h-oidev term of the quantum (15j)-symbol is given by:”

The tetrahedron contribution to the volume is the derivative of ln((6j)^) with respect to A.

“It is much more complicated to find the tetrahedron contribution to the volume, that is, the derivative of ln((6j)^) with respect to A.”

The action (38) can be written as a difference of two copies of Chern-Simons action (57).

“It is known, however, that the action (38) can be written as a difference of two copies of Chern-Simons action S CS (A) = ^-J TrfAAdA + ^AAAAA). (57 )”

Apparently there is no term proportional to h^2 = A in this expansion.

“Thus, apparently there is no term proportional to fr 2 = A in this expansion.”

If E is non-degenerate, equations (44) are satisfied iff E is the self-dual part of a decomposable 2-form, i.e., there exists a tetrad e^I such that E^i = ±(e^0 wedge e^1 + ...).

“In the case E is non degenerate, i.e., the right hand side of (44) is non zero, these equations are satisfied if and only if E is the self-dual part of a decomposable 2-form, i.e., if and only if there exists a tetrad field e 7 , / = 0,1,2,3 such that E l = ±(e 0 A e 1 + ^e^e- 7 A e k ).”

The tetrahedron contribution to the volume is a function of the spins labelling the edges of each tetrahedron.

“It is given by a function that depends on the spins labelling the edges of each tetrahedron.”

The vertex contribution to the volume is constant for each vertex.

“The contribution from the vertices is somewhat trivial -it is constant for each vertex.”

Taking E non-degenerate and right-handed, and substituting the spin connection for A in (38), yields the Euclidean Einstein-Hilbert action (41).

“Taking the E field to be non-degenerate and”

• Also stated in: § 3.3 3D BF theory with the 'cosmological term' · plain

  “and substituting into (38) the spin connection instead of A, one gets the Euclidean Einstein-Hilbert action l -j M d i xJg{R-2k). (41 )”

The expectation value of the volume is given by the derivative of amplitude (6) with respect to (-iA) evaluated at A=0.

“Indeed, the expectation value of the volume is given simply by the derivative of the amplitude (6) with respect to (-iA) evaluated at A = 0:”

Using gauge invariance and the property that an su(2) symmetric traceless tensor is characterized up to gauge transformation by its square, equation (122) is equivalent to (124).

“Using the fact that Z(J) and, therefore, Z(J) are gauge invariant and the property that an su(2) symmetric traceless tensor £2 U is totally characterized up to gauge transformation by its square fl^Qij we see that the preceding equation is equivalent to”

The requirement that (98) equals the volume of t fixes a = 1/(16·6).

“The requirement that (98) is equal to the volume of t fixes the parameter a to be 1/(16 • 6).”

The Reisenberger model uses the same simplest regularization of constraints, taking into account only the simplest types of intersections of wedges.

“However, the Reisenberger model, for example, uses exactly the same way of regularizing the constraints of the theory, for, using the terminology of our approach, it takes into account only the simplest types of intersections of wedges when regularizing the constraint part of the action.”

One difference from Reisenberger's model is that the derivative operators in this result are commutative, while Reisenberger uses non-commutative right and left invariant vector fields.

“The first one is the fact that the derivative operators appearing in our result are the commutative ones, while Reisenberger uses the non-commutative right and left invariant vector fields.”

Another difference is the presence of factors of P(J) in the generating functional.

“The second difference is due to the presence of the factors of P{ J) in the generating functional.”

In three dimensions, evaluating the interaction term on the distributional E field is not well-defined because contributions come from wedge intersections and the E field is distributional.

“Unfortunately, in the case of three dimensions the result is not well-defined as in the case of 2D Yang-Mills theory.Indeed, the E field is concentrated along the wedges (see Fig. 7), and non-trivial contributions to the integral (94) come from the points where wedges intersect.”

The function P(J)j, which is crucial in e.g. 2D Yang-Mills theory, is not accounted for by the Reisenberger and Barrett-Crane models.

“Also, the presence of the function P(J)j which plays a crucial role in, for example, 2D Yang-Mills theory, is not accounted for by the Reisenberger and Barrett-Crane models.”

In 4D, the approach did not reproduce correctly the contribution to the 4-volume coming from tetrahedra.

“As in 3D our approach did not reproduce correctly the terms corresponding to edges, in 4D we did not reproduce correctly the contribution to the 4volume coming from tetrahedra.”

The parameter a is fixed so that (113) equals (3/2)V_h when X_w is related to a geometrical 4-simplex.

“Thus, we will fix a in such a way that (113) is equal to (3/2)1^ when X l w can be related to the quantities characterizing a geometrical 4-simplex.”

The approach does not reproduce all terms appearing in (67).

“Note, however, that we do not get all of the terms appearing in (67).”

The solution of (124) can be written as equation (125).

“The solution of the later can be written as”

The first A-order term of the transition amplitude is given by equation (116).

“Thus, the first A-order term of the transition amplitude is given by:”

The trivalent grasping term in (67) exactly matches result (105), including numerical coefficient and sign.

“We will now show that the most complicated term in that expression -the term that involves trivalent graspings -exactly matches our result (105), including the numerical coefficient and the sign.”

After resolving ambiguities for the considered intersections, the result exactly matches analogous terms of usual state sum models, including numerical factors.

“However, the corresponding ambiguities can be successfully resolved for these types of intersections by using geometrical considerations, and, after the ambiguities are eliminated, the result exactly matches the analogous terms of the usual state sum models, including the matching of the numerical factors.”

For any triple, integral (97) equals a·sign(w1,w2,w3), with a independent of the triple.

“For any triple of wedges wi ^ W2 / ^3, the integral (97) is equal to a • sign(wi, W2, ttfs), where a is a coefficient that is independent on a triple, and signal, W2, w^) is plus or minus one depending on the orientation of the form dux A du2 A du-s.”

The expression (103) equals 12V, where V is the volume of the tetrahedron.

“Thus, (103) is equal to 12^, where V is the volume of t.”

In three dimensions, only partial agreement was reached between the approach's transition amplitudes for BF theory with a cosmological term and the corresponding state-sum models.

“and in the 3D case only a partial agreement was reached.”

The simplest regularization of the interaction (volume) term does not account for all terms obtained from the corresponding exact models.

“On examples of 3D and 4D BF theories with cosmological term we have seen that the simplest regularization of the”

The approach prescribes commuting variational derivatives with respect to the current instead of the non-commuting vector fields used by Reisenberger.

“Thus, our approach prescribes to use the commuting variational derivatives with respect to the current instead of non-commuting vector fields used by Reisenberger [3].”

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The approach gives the correct expression for the 2D Yang-Mills partition function apart from a factor exp(-e^2 p/4).

“Thus, our approach gives the correct expression (50) for the partition function of the Yang-Mills theory, apart from the factor exp(-e 2 p/4).”

In four dimensions, the approach's transition amplitudes for BF theory with a cosmological term agree with the corresponding state-sum models except for terms related to framing.

“In the case of 4D theory, we found agreement (apart from the terms that have to do with the framing)”

When E is the selfdual part of the wedge product of two tetrads, the integral (109) equals V_h, the volume of h with respect to the metric defined by the tetrad.

“when E in (109) is equal to the selfdual part of the wedge product of two copies of the frame field:”

• Also stated in: § Eab -^ab-i · plain

  “where Vh is the volume of h with respect to the metric defined by 6^.”

The state-sum formulation of topological BF theory can be obtained using distributional fields associated with a two-dimensional cellular complex.

“We have seen that the state sum formulation of the topological BF theory can be obtained by using distributional fields associated with a 2-dimensional cellular complex.”

Comparing (167) and (115) fixes a = 1/(30·4).

“A comparison of (167), (115) fixes the value of a to be a = 1/30 • 4.”

Using the correspondence (117), equation (116) agrees with result (81), including numerical coefficient and sign.

“Using this correspondence we see that (116) agree with the result (81).As in the 3D case, the matching includes the numerical coefficient and the sign of the expressions.”

The approach reproduces to some extent the transition amplitudes given by the Reisenberger and Barrett-Crane models.

“However, as we have seen on the example of the Reisenberger model, and as one can check for the case of Barrett-Crane model, our approach to some extent reproduces the transition amplitudes given by these models.”

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The approach accounts for the most intricate part of the 3D and 4D transition amplitudes at first order in A, reproducing the numerical coefficients correctly.

“Note, however, that our approach does account for the most intricate part of the 3D and 4D transition amplitudes in the first A-order, reproducing correctly even the numerical coefficients.”

In two dimensions, the transition amplitude at J=0 gives an expression involving the Euler characteristic of M, recognizable as the volume of flat connections modulo gauge transformations expressed via the Riemann zeta function.

“The corresponding two dimensional model is somewhat trivial: the transition amplitude is given by (25) at J = 0, which gives ^(dim^), (85) i where (M) is the Euler characteristics of M. One can recognize in this expression the volume of the space of flat connections on M, modulo gauge transformations expressed in term of Riemann zeta function (see, for exampie, [23]).”

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No geometrical arguments are currently available to fix the ambiguity in coefficients of these additional terms.

“Although the structure of terms appearing this way is clear, at the present state of the development of our approach, no geometrical arguments is available to fix the ambiguity in the coefficients in front of such terms.”

For 2D Yang-Mills, evaluating the interaction term on the distributional E field yields equation (89), with p_w the area of wedge w.

“Calculating this on the configuration of E given by (86), with each E w being constant along the wedge w, we get”

The two-dimensional Yang-Mills partition function depends only on the topological properties of the manifold and on the total area through the measure.

“Thus, as we indicated in the argument of YM, the partition function depends only on the topological properties of M. Also, the dependence on measure d”

Distributional solutions of this kind exist for a large class of theories.

“Solutions of this type can be shown to exist for a large class of theories.”

In those solutions, E is zero except in a tubular neighborhood of a two-dimensional surface.

“In these solutions the E field is zero except in a tubular neighborhood of a two-dimensional surface in spacetime.”

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The phase space of the theories treated is the same as that of Yang-Mills theory.

“The phase space of all theories we work with, as it can be seen from the above action functional, is that of Yang-Mills theory.”

The electric field operator in the canonical quantum theory is distributional.

“A characteristic feature of the quantum theory constructed with such a space of states is the distributional nature of the electric field operator.”

For some known state sum models, the precise link to the corresponding classical theory is not established.

“However, a precise relation between the classical theory and the corresponding state sum model is not known for some of these models.”

The confining phase is the phase in which monopole solutions dominate the path integral.

“In fact, the confining phase of the theory is the one in which monopole solutions dominate the path integral [20].”

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Baez showed that for gravity there are classical solutions in which E is distributional and concentrated on two-dimensional surfaces.

“As it was shown by Baez [17] for the case of gravity, there exist solutions of the classical equations of motion in which the E field is distributional and concentrated along two-dimensional surfaces in spacetime.”

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The monopole analogy suggests that these distributional solutions should be included in the path integral.

“Thus, the analogy with monopole solutions suggests that one should include these distributional solutions when taking the path integral.”

The history of electric field configurations can be regarded as a collection of two-dimensional surfaces in spacetime.

“This suggests that one should think of the”

In Yang-Mills theory, monopole solutions are distributional along lines.

“In Dirac's way [19] of treating the monopole, it is described by a configuration of fields in which magnetic field is distributional along certain lines.”

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The authors' method can construct a spin foam or state sum model for a theory from that theory's classical action.

“We propose a new systematic approach that allows one to derive the spin foam (state sum) model of a theory starting from the corresponding classical action functional.”

The method covers any theory whose action equals BF theory's action plus a functional of the B field.

“It can be applied to any theory whose action can be written as that of the BF theory plus a functional of the B field.”

In those solutions, away from the surface the connection is either arbitrary or flat, depending on the Lagrangian.

“The connection field away from the surface then must be either arbitrary or flat, depending on the detailed form of the Lagrangian of the theory.”

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There may be a phase in which the path integral is dominated by Baez's distributional solutions.

“Thus, there may exist a phase in a theory of the type we study where the path integral is dominated by the distributional solutions found by Baez.”

These states are exactly those found in the canonical approach.

“These are exactly the states one finds in the canonical approach to quantum heory, as we explained in the introduction.”

The result of (12) must be invariant under the discretized gauge transformations acting on Z_w and J_w.

“The result of ( 12) must be invariant under this gauge transformations.”

For any theory of the considered type, the transition amplitude is a formal power series in variational derivatives with respect to the current.

“Then the transition amplitude for any theory of the type we consider is given as a formal power series in variational derivatives 5/8 J with respect to the current field.”

Wedge variables arise naturally when implementing gauge symmetry.

“Another motivation for using them is, as we explain below, that the wedge variables appear naturally when one tries to implement the gauge symmetry in the theory.”

The action (38) is BF theory in 3D with E as B and an additional cosmological term.

“Thus, the action (38) is that of BF theory in 3d, with E field playing the role of 5, and with an additional”

This 3D theory is related to gravity: from the one-form E one can construct a real Euclidean-signature metric g_ab = -1/2 Tr(E_a E_b).

“This theory is related to gravity in 3D as follows.Having the one-form E, one can construct from it a real metric of Euclidean signature gab = -^Tr{E a E b ). (39 )”

The partition function calculation can be reduced to the vacuum-vacuum transition amplitude of a different theory with action (37).

“The problem of calculation of the partition function can be reduced to the problem of calculation of the vacuum-vacuum transition amplitude of a somewhat different theory, that is of the path integral of exp (i x action), with the action given by / Tr(EF)-i^ f d^{E 2 ).(37)”

For such theories, approximation (5) should improve as the triangulation becomes finer and dual faces smaller.

“However, even for such theories, one would expect the approximation (5) become better as the triangulation of the manifold becomes finer and the dual faces become smaller.”

The result (13) can be checked by integrating it with the disk partition function over dg to reproduce the punctured-sphere partition function of [23].

“The result (13) can then be checked by taking the partition function on the disk (equal to d(g), where g is the holonomy along the boundary of the disk), and integrating it with (13) over dg.This must give the partition function on a punctured sphere, and indeed reproduces the result given in [23].”

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The relation to 2D BF theory specifies the measure dX_w.

“The following relation to 2D BF theory will also serve the purpose of specifying this measure.”

Approximation (5) is harder to justify for theories whose equations of motion do not force flat connections, such as BF with cosmological term or gravity.

“It is harder to justify this approximation for theories, for which the classical equations of motion do not imply the connection to be flat, as is the case, for example, for BF theory with cosmological term or for gravity.”

The state can depend on A only through holonomies along the edges.

“This means that the state $ can depend on the values of A only as a function of the holonomies of the connection along the edges.”

The integral (12) can be related to the 2D BF theory path integral (16) on a wedge.

“To find the result of the integral (12) we relate it to a more complicated integral, for which a precise result is known from 2D BF theory calculation.”

2D Yang-Mills is described by the BF-like action (34).

“Yang-Mills theory in 2D is described by the following BF-like action / Tr(EF) + e -f dfiTriE 2 ). (34 )”

The densitized E field becomes the variational derivative operator with respect to A.

“The densitized vector field E (that is dual to E) becomes the operator of variational derivative with respect to A.”

The discretization used is consistent with the canonical approach.

“The discretization we consider is consistent with the results of the canonical approach.”

With Z_w defined this way, the discretized action is gauge invariant.

“With Z w defined this way the discretized action is gauge invariant.”

Flat connections play the dominant role in BF theory.

“Thus, the flat connections play the dominant role in BF theory.”

On a spatial hypersurface, E can be described as a collection of flux lines of non-abelian electric field.

“Indeed, the restriction of E on a spatial hypersurface can be described as a collection of flux lines of non-abelian electric field.”

The equal-norm constraint on X_w need not be imposed explicitly because integrating over group elements automatically enforces it.

“However, as we shall see, there is no need to impose these constraints: one can integrate over all X w independently, but later, when one integrates over the group elements, only the contributions that come from X w of an equal norm will survive.”

• Also stated in: § Jw · plain

  “there is no need to impose the constraint that length of all wedge variables X w belonging to one and the same dual face are equal: this constraint gets imposed automatically when the integration over the group elements is performed.”

Approximation (5) is natural from the perspective of two-dimensional BF theory.

“As we shall see, the approximation ( 5) is quite natural from the point of view of 2D BF theory.”

Approximation (5) is justified for BF theory because only connections close to flat matter.

“Thus, this approximation is certainly justified for BF theory, where we expect only connections close to flat to matter in the quantum theory.”

In two dimensions, because dual faces cover the manifold, the E field can be replaced by a field constant on each wedge and equal to X_w.

“However, since the spacetime is two-dimensional, the dual faces cover the manifold, and we just have to replace the E field by a field constant on each wedge, and equal to the Lie algebra element X w .”

Even for BF theory, wedge variables are needed to reproduce a triangulation-independent state sum model when dual face topology is not a disc.

“It is interesting, however, that, even in the case of BF theory one has to use the wedge variables if one wants to reproduce a triangulation-independent state sum model for the case the topology of a is different from that of a disc.”

The spin labelling a wedge in (15) has the meaning of the length of the corresponding Lie algebra element X.

“As one can see from the formula (137) for the inverse Fourier transform, the spin labelling a wedge in (15) has the meaning of the length of the corresponding Lie algebra element X.”

Approximation (5) is good when Z_w is close to zero in the Lie algebra.

“The approximation (5) is good for Z w close to zero element in the Lie algebra.”

For E to vanish except along graph edges, the state must depend on the connection only along those edges.

“In order for E in a state”

The state sums (46) and (47) are realizations of the integral over connections with integrand delta at F=0.

“The state sums (46),(47) are in certain precise sense realizations of the integral over connections with the integrand being the delta-function at F = 0.”

Each edge of the triangulation carries an intrinsic volume that depends on its spin.

“Again, this implies that each edge of the triangulation A carries an intrinsic volume that depends on its spin.”

The E field in (38) has the interpretation of the triad field.

“Thus, the E field in (38) has the interpretation of the triad field.”

The expectation value of the volume of M in BF theory is proportional to the first-order term in A in the decomposition of (55).

“Thus, the expectation value of the volume of M in BF theory is proportional to the first order term in A in the decomposition of (55).”

Reisenberger proposed a state sum model corresponding to the selfdual Plebanski action.

“Reisenberger [3] has proposed a state sum model corresponding to the selfdual Plebanski action (43).”

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If M has a boundary, the Crane-Yetter invariant is proportional to the Chern-Simons transition amplitude CS(k, ∂M).

“It has been shown (see e.g.[25]) that if M is a manifold with boundary then CY(g, M) is proportional to the Chern-Simons transition amplitude CS(fc,dM) introduced in the previous subsection:”

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In four dimensions, the parameter h equals the cosmological constant A.

“Thus, unlike in the case of 3-dimensions, we have now h = A.”

The result (67) matches the Heegard splitting picture, with volume concentrated in vertices, edges, and tetrahedra.

“Our result (67) provides one with exactly the same picture: the volume of M is concentrated in vertices (balls of the Heegard splitting), edges (cylinders), and tetrahedra (4-holed spheres).”

Vol(A,j) has the interpretation of the volume of a labelled triangulation.

“This shows that Vol(A,j) indeed has the interpretation of the volume of a labelled triangulation.”

Thus (43) is the BF action in four dimensions with an additional simplicity constraint.

“Thus, (43) is the BF action in four dimensions, with an additional”

The Heegard splitting of a triangulated manifold M yields building blocks: balls, cylinders, and spheres with four discs removed, each carrying part of the original volume.

“Each of this manifolds carries a part of the original volume of M.”

The Turaev-Viro invariant (55) can be interpreted as the vacuum-vacuum transition amplitude of the theory defined by (38).

“The construction that interprets the Turaev-Viro invariant (55) as the vacuum-vacuum transition amplitude of the theory defined by (38) is as follows.”

The Yang-Mills partition function calculation can be reduced to the path integral (49).

“As we discussed above, the problem of calculation of the Yang-Mills partition function can be reduced to the problem of calculation of the path integral Z YM (pe 2 : M) = fvAVEexp li f Tr(EF) + y / dfiTr(E 2 ) ) , (49 )”

The first-order term in the decomposition of (55) in powers of A is proportional to the expectation value of spacetime volume in BF theory at A=0.

“To compare in Sec. 5 the spin foam model obtained via our techniques with the Turaev-Viro model, we will need the first-order term in the decomposition of (55) in the power series in A. This is proportional to the expectation value of the spacetime volume in BF theory at A = 0.”

The Reisenberger and Barrett-Crane models are invented rather than derived from the Einstein-Hilbert action or an equivalent.

“The major problem with both of them is that they are rather invented then derived as a result of some systematic procedure starting from the Einstein-Hilbert action or any equivalent of it.”

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The additional terms are exactly the types needed to account for other terms appearing in the first A-order of the Turaev-Viro model.

“In fact, as one can see, these are exactly the types of terms that are needed to account for the other terms appearing in the first A-order in the Turaev-Viro model.”

The norm of the self-dual part of each bivector equals the squared area of the corresponding face.

“As is shown in the Appendix E (see (166)), the norm (obtained using the usual flat metric in IR 3 ) of the self-dual part of each bivector is equal to the squared area of the corresponding face.”

The partition function of 2D Yang-Mills may depend on the regularization procedure, with different schemes differing by factors of exp(-a e^2 p).

“That is, the result for the partition function of 2D Yang-Mills may depend on a regularization procedure that was used to calculate it, the two different schemes giving results that may differ, in particular, by factors of exp(-ae 2 p), where a is some coefficient.See, for instance, [6] for a more detailed discussion of the standard renormalizations.”

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The ambiguity in the 3D integral (95) can be resolved for intersections of three different wedges at the centers of tetrahedra using geometrical considerations.

“In this paper we show how the ambiguity in (95) can be resolved for the simplest, and most important type of intersections: intersections of three different wedges at the centers of tetrahedra.As we shall see, for intersections of this type there is a simple way to resolve the ambiguity in (95) using geometrical considerations.”

The constraint equations of the known spin foam models have the meaning of differential equations that the generating functional must satisfy.

“We have also seen that the constraint equations of the known spin foam models have the meaning of differential equations that must be satisfied by the generating functional.”

The discrepancy between the model obtained and the usual state sum model is understood by comparing the two expressions for spacetime volume: Turaev-Viro associates volume to tetrahedra, edges, and vertices, while this approach accounts only for the tetrahedron part.

“The discrepancy we find between the model we obtain and the usual state sum model can be understood by comparing the two different expressions for the spacetime volume: the one obtained within our approach and the one obtained within the Turaev-Viro model, see (67).The Turaev-Viro model tells us that we must associate the spacetime volume not only to tetrahedra of the triangulation, but also to”

The reason only the tetrahedron part is accounted for is that the evaluation of the cosmological term on the distributional E field included only intersections of different wedges at tetrahedron centers.

“The reason for this was that, when evaluating the cosmological term of the action on the distributional E field, we took only the terms which came from intersections of different wedges at the centers of tetrahedra.”

The authors put forward a systematic method for deriving known spin foam state-sum models from their corresponding classical action principles.

“We have proposed such a procedure, motivated by certain results of the loop approach to quantum gravity.”

• AssumptionRests on an assumption: Known spin foam state-sum models can be obtained from a systematic procedure starting from the corresponding classical action principle.

The approach gives an interpretation of the evaluation of spin networks satisfying the flat-connection constraint equations as taking J=0 at the end of a transition amplitude calculation.

“In our approach this evaluation has an obvious meaning of taking J = 0 at the end of the calculation of a transition amplitude.”

The partition function obtained by this method differs from the Migdal-Witten model only by a standard renormalization factor.

“Thus, the partition function obtained by our method differs from that of the Migdal-Witten model just by a standard renormalization factor.”

The interpretation of spin j as edge length for large j is required to reproduce the Regge calculus version of the Einstein-Hilbert action in the Ponzano-Regge model.

“Another motivation for interpreting the spin j as the length of an edge (for large j) is that it is exactly this interpretation that must be used to reproduce correctly the Regge calculus version of Einstein-Hilbert action in the Ponzano-Regge model of quantum gravity [7].”

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The authors' experience with BF theories indicates that the Reisenberger model's regularization cannot be correct.

“Our experience with the BF theories tells us that this cannot be correct.”

The missing contributions in the 3D and 4D cases can be interpreted as singular contributions that arise when the volume operator is regularized.

“We have also seen, both in the cases of 3D and 4D, that the missing contributions can be interpreted as coming from singular contributions arising when one regularizes the volume operator.”

The procedure proposed in the paper is still far from achieving the goal of deriving spin foam models from action functionals.

“However, the procedure proposed in this paper is still far from being able to claim to have achieved this goal.”

The existing models of quantum gravity in four dimensions are too oversimplified.

“Our results also led us to the conclusion that the existing models of quantum gravity in 4D are too oversimplified.”

Reisenberger's model modifies the SU(2) Crane-Yetter model to implement the constraints (44).

“The main idea of his construction is to modify the SU(2) Crane-Yetter model in such a way that the constraints (44) are implemented.”

The universal invariant evaluation U_{G,C}(Z(L)) equals the Reshetikhin-Turaev evaluation of the link L associated with the quantum group U_q(G).

“One important property of the universal invariant is that UG,C{Z(L)) is equal to the Reshetikhin-Turaev evaluation of the link L associated with the quantum group U q (G) [31,30,32]”

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The volume of a 4-simplex can be expressed as a wedge product of any two bivectors corresponding to faces that do not share an edge.

“The volume of h can be expressed as a wedge product of any two of the bivectors (159) corresponding to faces that do not share an edge.”

There exists an expression for the volume of a 4-simplex that involves only the self-dual parts of bivectors.

“There exists an expression for the volume of h that involves only the self-dual parts of bivectors (159):”

The vacuum-vacuum transition amplitude of the theory is given by the expression in equation (55).

“The vacuum-vacuum transition amplitude of the theory is then given by the following expression (see, for example, [25]):”

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