D-Theory: A Quest for Nature's Regularization

Nuclear Physics B (Proc. Suppl.) 153 (2006) 336–347 www.elsevierphysics.com

D-Theory: A Quest for Nature’s Regularization U.-J. Wiesea a

Institute of Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

D-theory provides a nonperturbative regularization for quantum field theory in which classical fields emerge dynamically from the dimensional reduction of discrete variables. The D-theory variables — quantum spins and quantum links — are possible candidates for the physical degrees of freedom that Nature may have chosen to regularize the standard model of particle physics at ultrashort distances. D-theory provides a natural solution for the nonperturbative hierarchy problem of 4-dimensional fermions with a chiral symmetry. If D-theory resembles Nature’s physical regularization, the parameters of the standard model are analogous to material specific lowenergy constants of a piece of condensed matter.

1. Introduction What are the truly fundamental physical degrees of freedom that underlie particle physics? The identification of the ultimate hardware on which the basic laws of Nature are implemented may or may not be within reach of physics in the foreseeable future. No matter if there are superstrings or some tiny wheels turning around at the Planck scale, Nature must have found a concrete way to regularize gravity as well as the standard model at ultrashort distances. Dirac continued to point out that he was unsatisfied with the rather formal procedures of removing singularities in the perturbative treatment of QED, and he pointed out the need for a physical regularization [1]. Obviously, Nature’s regularization must be concrete and nonperturbative. Man-made mathematical regularizations like analytic continuation in the space-time dimension are elegant and convenient, but they define the theory only in perturbation theory. In this talk, it is argued that not only strings and branes but also discrete variables — namely quantum spins and their gauge analogs, quantum links — are possible candidates for Nature’s most fundamental degrees of freedom. D-theory is a nonperturbative formulation of field theory in which classical fields emerge from the collective dynamics of discrete quantum variables which undergo dimensional reduction. D-theory was de0920-5632/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2006.01.027

veloped some time ago as a discrete approach to U (1) and SU (2) pure gauge theories [2], extended to SU (N ) gauge theories and full QCD in [3–5], and applied to a variety of problems in [6–11]. In this talk, D-theory is emphasized as a potential framework for a truly fundamental theory. In order to motivate the D-theory approach, we first like to discuss fundamental physics from a nonperturbative point of view. 2. A Nonperturbative View at Fundamental Physics Why is gravity so weak? This is one of many hierarchy problems in physics. As Wilczek has pointed out, the feebleness of gravity is intimately related to asymptotic freedom [12]. The ratio of the gravitational and electrostatic forces between two protons at some large distance (where G is Newton’s constant and e is the proton’s electric charge), Gm2 Fg = 2 ≈ 137 Fe e




m MP

2

≈ 10−36 ,

(1)

is determined by the ratio of the √ proton mass m and the Planck scale MP = 1/ G. In order to understand why gravity is so weak, we must hence figure out why the proton is so light compared to the Planck scale. Asymptotic freedom solves part of this hierarchy problem because it guarantees that the QCD scale is naturally much

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smaller than the ultimate ultraviolet cut-off, by a factor that is exponentially large in the inverse QCD gauge coupling 1/g 2 (evaluated at the Planck scale), i.e.
 m 24π 2 ∝ exp − . (2) MP (11Nc − 2Nf )g 2 Here Nc = 3 is the number of colors and Nf is the number of quark flavors. For simplicity we have assumed that there are no other colored objects (such as, for example, GUT gauge bosons) up to the Planck scale. While it is easy to incorporate such degrees of freedom, they have no impact on Wilczek’s argument (at least as long as they do not destroy asymptotic freedom). The crucial point is that the value of g at the Planck scale need not be fine-tuned. Instead, due to the exponential factor resulting from asymptotic freedom, a large hierarchy such as m/MP ≈ 10−19 arises naturally. Asymptotic freedom explains naturally why non-Abelian gauge fields, in this case gluons, exist far below the ultimate ultraviolet cut-off. This alone would suggest that ordinary matter should consist of glueballs. However, protons and neutrons also consist of light quarks. So, why are there light fermions in Nature? This question is related to another deep hierarchy problem which has, in fact, been solved during the past 15 years. However, it seems that neither the existence of the problem nor its solution have been widely recognized [13]. Obviously, the existence of light fermions is related to chiral symmetry. Most of the time chiral symmetry is discussed in the perturbative context of continuum field theory. In particular, the global chiral symmetry of massless QCD is usually taken for granted because it can easily be maintained in continuum regularization schemes. A continuum field theorist could “explain” the presence of (almost) massless fermions in Nature by the existence of (an approximate) chiral symmetry which protects the quark masses from running to the cut-off scale. As will be stressed here, this perturbative point of view of the problem is somewhat narrow, and may even prevent us from drawing some interesting conclusions about the physics at very high energy scales.

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Beyond perturbation theory, e.g. in Wilson’s lattice field theory [14], due to fermion doubling [15,16] chiral symmetry has posed severe problems for many years. In particular, using Wilson fermions, i.e. removing the doubler fermions by breaking chiral symmetry explicitly, causes a severe nonperturbative hierarchy problem for fermions. Without unnatural fine-tuning of the bare fermion mass it is then impossible to obtain light fermions. Some time ago, lattice field theorists could thus not explain naturally why there are light fermions in Nature. This problem has sometimes been viewed as a deficiency of the lattice regularization. However, beyond perturbation theory there is indeed a deep hierarchy problem for fermions which manifests itself most clearly on the lattice, but which still affects particle physics in general. The fact that the problem is not easily recognizable in perturbation theory may even be viewed as a drawback of continuum regularization schemes. Remarkably, the hierarchy problem of the nonperturbative regularization of chiral symmetry has found an elegant solution in terms of Kaplan’s domain wall fermions [17]. As Rubakov and Shaposhnikov first noticed, massless 4-d fermions arise naturally as states localized on a domain wall embedded in a 5-d space-time [18]. Kaplan showed that this mechanism also works on the lattice, while fermion doublers are still removed by a 5-d Wilson term. Narayanan and Neuberger’s closely related overlap fermions [19] are also deeply related to the physics of an extra dimension. A sufficient condition for the existence of a lattice chiral symmetry [20] is the GinspargWilson relation [21]. Domain wall or overlap fermions obey the Ginsparg-Wilson relation naturally. On the other hand, Hasenfratz and Niedermayer’s classically perfect 4-d lattice fermion actions [22], although extremely useful for solving QCD numerically, obey this relation as a result of a very elaborate form of fine-tuning. Hence, solving the nonperturbative hierarchy problem of fermions naturally, i.e. without fine-tuning, seems to require at least one extra dimension. Without invoking extra dimensions, at a nonperturbative level we presently do not understand how fermions can be naturally light. The existence

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of light fermions in Nature may thus be a concrete hint to the physical reality of extra dimensions. In particular, we don’t need string theory or other physics beyond the standard model to motivate extra dimensions. The mere existence of light fermions in Nature is evidence already. This important hint from nonperturbative physics is indeed easily missed when one considers chiral symmetry only in a perturbative context. Why is the weak scale so much smaller than the GUT or Planck scale? This is the generally appreciated gauge hierarchy problem of the standard model. In contrast to the nonperturbative hierarchy problem of chiral symmetry, the gauge hierarchy problem manifests itself already in perturbation theory. The early attempts to solve this hierarchy problem by using the asymptotic freedom of a technicolor gauge theory unfortunately faced severe problems with flavor-changing neutral currents, and thus do not seem to work. The presently most popular potential solution of the gauge hierarchy problem relies on supersymmetry. Fermionic superpartners of the Higgs field would cancel the quadratic divergence in loop diagrams of the scalar self-energy and would thus eliminate the need for unnatural fine-tuning of the bare mass of the scalar field in each order of the perturbative expansion. However, from a nonperturbative point of view this “solution” is not yet satisfactory. Beyond perturbation theory, e.g. on the lattice, a priori supersymmetry is as undefined as chiral symmetry was before Kaplan constructed lattice domain wall fermions. First of all, lattice scalar field theory suffers from the same hierarchy problem as the continuum theory. Hence, without unnatural fine-tuning of the bare mass, the physical mass as well as the vacuum expectation value of the scalar field remain at the ultraviolet cut-off. In other words, the scalar correlation length is just a few lattice spacings, and one is not close to a continuum limit. When Wilson fermions are added, they suffer from their own nonperturbative hierarchy problem. Supersymmetry is still expected to emerge in the continuum limit which, however, can only be reached by unnatural fine-tuning of both the bare scalar and fermion masses. Replacing Wilson fermions

by domain wall fermions alone does not help because fine-tuning would still be required for obtaining light scalars. Of course, as long as the nonperturbative construction of supersymmetry itself requires fine-tuning, it cannot solve the hierarchy problem. In the worst case, the supersymmetric extension of the standard model may just be a perturbative illusion which does not arise naturally, i.e. without fine-tuning, in a nonperturbative context. Unfortunately, in contrast to chiral symmetry, Nature has not yet provided us with evidence for supersymmetry. While this may well change in the near future, one can presently not be sure that supersymmetric extensions of the standard model even exist naturally beyond perturbation theory. If they don’t, they can obviously not solve the gauge hierarchy problem or help to unify the gauge couplings at the GUT scale. Of course, there is also no reason to be too pessimistic about the existence of supersymmetry beyond perturbation theory. There are even very interesting recent lattice developments [23], and a number of supersymmetric theories have already been constructed nonperturbatively. However, it still remains to be seen if these developments will lead to a nonperturbative solution of the gauge hierarchy problem. The presently most popular view seems to be that string theory or M-theory may provide the correct extension of the standard model to very high energies. A priori, also string theory is not defined satisfactorily beyond perturbation theory. While there are very interesting attempts to base string theory on a nonperturbatively defined Mtheory, at present the relation of M-theory to the standard model seems rather indirect. The most concrete regularization of standard model physics is presently provided by Wilson’s lattice field theory. It should be stressed that lattice field theory is not an approximation to any pre-existing nonperturbatively well-defined theory in the continuum. Instead, the lattice regularization defines the (otherwise only perturbatively defined) theory beyond perturbation theory. The corresponding continuum theory emerges at a critical point of the 4-d lattice model. Then the physical correlation length diverges in units of the lattice spacing, and the theory becomes insensi-

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tive to irrelevant details at the cut-off scale. Using lattice perturbation theory, it has been shown analytically that, after removing the cut-off, lattice field theory defines the same perturbative theory as continuum regularization schemes. Of course, in contrast to the lattice, those schemes do not define the theory nonperturbatively. Due to the fermion doubling problem, for a long time it seemed that chiral gauge theories cannot be regularized on the lattice. After the Ginsparg-Wilson relation was better understood, this situation has changed drastically. In particular, using his lattice variant of chiral symmetry [20], L¨ uscher was able to construct lattice chiral gauge theories that are manifestly gauge invariant and still free of doubler fermions [24]. Remarkably, in order to be consistent at the quantum level, the lattice theory must satisfy the same anomaly cancellation conditions as the continuum theory [24,25]. L¨ uscher’s construction applies to both Abelian and non-Abelian chiral gauge theories. Even perturbation theory can benefit from these developments because the ambiguities related to the regularization of γ5 in multi-loop diagrams are uniquely fixed in the lattice construction. When implemented in five dimensions with domain wall or overlap fermions, L¨ uscher’s construction solves the nonperturbative fermion hierarchy problem and explains without fine-tuning why there are light chiral fermions in Nature. Could Wilson’s lattice field theory be the shortdistance regularization that Nature has chosen for the standard model physics? At least, in contrast to string theory, it is sufficiently well defined that even a digital computer can “understand” it. However, while this is not logically impossible, it seems far too baroque to imagine that Wilson’s parallel transporter link variables are Nature’s truly fundamental degrees of freedom. Although they have the advantage of interacting nonperturbatively, Wilson’s link variables are obviously as man-made as, for example, dimensional regularization. Just like perturbative quantum field theory, Wilson’s lattice field theory still uses the concept of fundamental classical fields. It is difficult to imagine that classical fields, attached to each point in space, are Nature’s most fundamental microscopic degrees of

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freedom. After all, they represent a classical concept that is likely to break down at ultrashort distance scales. This view is consistent with string theory, in which at the Planck scale classical fields are replaced by strings and branes. However, even string theory itself needs a nonperturbative foundation like M-theory. One suggestion of this talk is to try to circumvent string theory by D-theory which could perhaps be more easily related to the standard model physics. D-theory provides a framework in which the familiar classical fields emerge naturally from discrete quantum variables that undergo dimensional reduction. This formulation provides a simple regularization for interesting field theories such as, for example, 2-d CP (N − 1) models or 4-d QCD. In the D-theory formulation of QCD a fifth dimension is not only needed to obtain naturally light quarks, but also to assemble 4-d gluons out of 5-d quantum links. 3. The Algebraic Structure of D-Theory In D-theory continuous classical fields are replaced by discrete quantum variables — quantum spins and their gauge analogs, quantum links. In the following the basic building blocks of Dtheory are introduced. 3.1. Real Vectors Let us discuss how an N -component unitvectors of an O(N ) model should be represented in D-theory. An important hint comes from the quantum XY model which has an SO(2) symmetry. The 2-component unit vector (s1 , s2 ) of the classical XY model is then replaced by the first two components of a quantum spin (S 1 , S 2 ) which indeed form a vector under SO(2). This has a natural generalization to higher N . Let us consider the (N +1)N/2 generators of SO(N +1). Among them, N generators S i (i ∈ {1, 2, ..., N }) transform as a vector under SO(N ) and the remaining N (N −1)/2 generators generate the subgroup SO(N ). In other words, in the subgroup decomposition SO(N + 1) ⊃ SO(N ) the adjoint representation of SO(N + 1) decomposes as {

N (N − 1) (N + 1)N }={ } ⊕ {N }. 2 2

(3)

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3.2. Real Matrices Spin models with an SO(N )L ⊗ SO(N )R symmetry are formulated in terms of classical real SO(N ) matrix fields. Similarly, in SO(N ) lattice gauge theory one deals with real valued classical parallel transporter matrices o which transform appropriately under SO(N )L ⊗ SO(N )R gauge transformations on the left and on the right. This symmetry is generated by N (N − 1) Hermitean operators. In D-theory, the real valued classical matrix o is replaced by an N × N matrix O whose elements are Hermitean operators. Altogether, this gives N (N − 1) + N 2 = N (2N − 1) generators — the total number of generators of SO(2N ). The corresponding subgroup decomposition SO(2N ) ⊃ SO(N )L ⊗ SO(N )R yields

ements of the matrix U are described by 2N 2 Hermitean generators — N 2 representing the real part and N 2 representing the imaginary part of the classical matrix u. Altogether, we thus have 2(N 2 − 1) + 1 + 2N 2 = 4N 2 − 1 generators — the number of generators of SU (2N ). The corresponding subgroup decomposition SU (2N ) ⊃ SU (N )L ⊗ SU (N )R ⊗ U (1) takes the form

N (N − 1) N (N − 1) , 1} ⊕ {1, } 2 2 ⊕ {N, N }. (4)

3.5. Symplectic, Symmetric, and AntiSymmetric Tensors The third main sequence of Lie groups (besides SO(N ) and SU (N )) are the symplectic groups Sp(N ). The group Sp(N ) is a subgroup of SU (2N ) whose elements Ω obey the additional constraint Ω∗ = JΩJ † . The real skew-symmetric 1. It is interesting to matrix J obeys J 2 = −1 ask how Sp(N ) gauge theories can be formulated in D-theory. In fact, this is completely analogous to the SO(N ) and SU (N ) cases. The Sp(N )L ⊗ Sp(N )R symmetry transformations are generated by N (2N + 1) Hermitean operators on the left and N (2N + 1) operators on the right. In D-theory, the (2N )2 elements of an Sp(N ) symplectic matrix are described by 4N 2 Hermitean operators. Altogether, we thus have 2N (2N + 1) + 4N 2 = 2N (4N + 1) generators — the number of generators of Sp(2N ). The corresponding subgroup decomposition Sp(2N ) ⊃ Sp(N )L ⊗ Sp(N )R takes the form

{N (2N − 1)} = {

3.3. Complex Vectors We have seen how to represent a real N component vector s in D-theory. It is simply replaced by an N -component vector of Hermitean generators S i of SO(N +1). In CP (N −1) models, classical N -component complex vectors z arise. We will now discuss their representation in Dtheory. The symmetry group of a CP (N − 1) model is U (N ) which has N 2 generators. In D-theory the complex components z i are represented by 2N Hermitean operators — N for the real and N for the imaginary parts. Hence, the total number of generators is N 2 +2N = (N +1)2 −1 — the number of generators of SU (N +1). In this case, the subgroup decomposition SU (N + 1) ⊃ SU (N ) ⊗ U (1) takes the form {(N + 1)2 − 1} = {N 2 − 1} ⊕ {1} ⊕ {N } ⊕ {N }.(5) 3.4. Complex Matrices Chiral spin models with a global SU (N )L ⊗ SU (N )R ⊗ U (1) symmetry as well as U (N ) and SU (N ) lattice gauge theories are formulated in terms of classical complex U (N ) matrix fields. The corresponding symmetry transformations are generated by 2(N 2 − 1) + 1 Hermitean operators. In D-theory a classical complex valued matrix u is replaced by a matrix U whose elements are non-commuting operators. The el-

{4N 2 − 1} = {N 2 − 1, 1} ⊕ {1, N 2 − 1} ⊕ {1, 1} ⊕ {N, N } ⊕ {N , N }.

(6)

The above structure is used in U (N ) and SU (N ) quantum link models in which the elements of Wilson’s classical parallel transporter matrices are replaced by non-commuting operators.

{2N (4N + 1)}={N (2N + 1), 1} ⊕{1, N (2N + 1)} ⊕ {2N, 2N }. (7) Other useful building blocks for D-theory models are symmetric (S T = S) and anti-symmetric (AT = −A) tensors which transform as S  = ΩSΩT , A = ΩAΩT ,

(8)

under SU (N ) transformations. In D-theory, the N (N +1)/2 elements of a complex symmetric ten-

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sor are represented by N (N + 1) Hermitean operators. In addition, there are again N 2 − 1 SU (N ) and one U (1) generators. Hence, altogether, there are N (N + 1) + N 2 = N (2N + 1) generators — namely those of Sp(N ). In this case, the subgroup decomposition is Sp(N ) ⊃ SU (N ) ⊗ U (1) and it takes the form N (N + 1) } {N (2N + 1)}={N 2 − 1} ⊕ {1} ⊕ { 2 N (N + 1) ⊕{ }. (9) 2 Similarly, the N (N − 1)/2 elements of a complex anti-symmetric tensor are represented by N (N − 1) Hermitean operators. Again, there are also N 2 − 1 generators of SU (N ) and one U (1) generator. In total, there are N (N − 1) + N 2 = N (2N − 1) generators — exactly those of SO(2N ). Now the subgroup decomposition SO(2N ) ⊃ SU (N ) ⊗ U (1) takes the form {N (2N − 1)}={N 2 − 1} ⊕ {1} ⊕ {

N (N − 1) } 2

N (N − 1) }. (10) 2 To summarize, in D-theory classical real and complex vectors s and z are replaced by vectors of operators S and Z which are embedded in SO(N + 1) and SU (N + 1) algebras, respectively. Similarly, classical real and complex valued matrices o and u are replaced by matrices O and U with operator valued elements which are embedded in the algebras of SO(2N ) and SU (2N ). In addition, 2N ×2N symplectic matrices, as well as N × N symmetric and anti-symmetric complex tensors are represented by the embedding algebras Sp(2N ), Sp(N ), and SO(2N ), respectively. In the next sections, we will use some of these basic building blocks to construct the D-theory formulation of QCD.

⊕{

4. U (N ) and SU (N ) Quantum Link Models Wilson’s formulation of lattice gauge theory uses classical complex SU (N ) parallel transporter link matrices ux,µ with an action  Tr(ux,µ ux+ˆµ,ν u†x+ˆν ,µ u†x,ν ). (11) S[u] = − x,µ=ν

The action is invariant under SU (N ) gauge transformations αx · λ)ux,µ exp(−i αx+ˆµ · λ). ux,µ = exp(i

(12)

In D-theory, the action is replaced by the action operator  † † Tr(Ux,µ Ux+ˆµ,ν Ux+ˆ (13) H=J ν ,µ Ux,ν ). x,µ=ν

Here the elements of the N × N quantum link operators Ux,µ consist of generators of SU (2N ). Gauge invariance now means that H commutes  x of gauge transforwith the local generators G mations at the site x, which obey [Gax , Gby ] = 2iδxy fabc Gcx .

(14)

Gauge covariance of a quantum link variable requires     y )Ux,µ  z) = exp(−i αy · G exp(i αz · G Ux,µ y

z

= exp(i αx · λ)Ux,µ exp(−i αx+ˆµ · λ), (15)   x ) is the unitary operator where x exp(i αx · G that represents a general gauge transformation in Hilbert space. The above equation implies the following commutation relation  x , Uy,µ ] = δx,y+ˆµ Uy,µλ − δx,y λUy,µ . [G

(16)

It is straightforward to show that this is satisfied when we write  x =  x,µ ),  x−ˆµ,µ + L G (17) (R µ

 x,µ and L  x,µ are generators of right and where R left gauge transformations of the link variable Ux,µ . The commutation relations of eq.(16) imply  x,µ , Uy,ν ] = δx,y δµν Ux,µλ, [R  x,µ , Uy,ν ] = −δx,y δµν λUx,µ . [L

(18)

Together with a U (1) generator Tx,µ the above operators form the link based algebra of SU (2N ). One finds [Tx,µ , Uy,ν ] = 2δx,y δµν Ux,µ ,

(19)

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which implies that Gx =

1 (Tx−ˆµ,µ − Tx,µ ) 2 µ

(20)

generates an additional U (1) gauge transformation, i.e.    = exp(−iαy Gy )Ux,µ exp(iαz Gz ) Ux,µ y

z

= exp(iαx )Ux,µ exp(−iαx+µ ).

(21)

Indeed the action operator of eq.(13) is also invariant under the additional U (1) gauge transformations and thus describes a U (N ) lattice gauge theory. In order to reduce the symmetry of the quantum link model from U (N ) to SU (N ) one breaks the additional U (1) gauge symmetry by adding the real part of the determinant of each link matrix to the action operator  † † Tr(Ux,µ Ux+ˆµ,ν Ux+ˆ H = J ν ,µ Ux,ν ) x,µ=ν

+ J





† (detUx,µ + detUx,µ ).

(22)

x,µ

5. Classical Fields from Dimensional Reduction of Discrete Variables Let us consider SU (N ) non-Abelian gauge theories, first in d = 4. The action operator of the corresponding quantum link model which is defined on a 4-d lattice, describes the evolution of the system in a fifth Euclidean direction. The partition function Z = Tr exp(−βH) can then be represented as a (4+1)-d path integral. Note that we have not included a projector on gauge invariant states, i.e. gauge variant states also propagate in the fifth direction. This means that we do not impose a Gauss law in the unphysical direction. Not imposing Gauss’ law implies A5 = 0 for the fifth component of the gauge potential. This is convenient because it leaves us with the correct field content after dimensional reduction. Of course, the physical Gauss law is properly imposed because the model does contain nontrivial Polyakov loops in the Euclidean time direction. Dimensional reduction in quantum link models works differently than for quantum spins. In

the spin case the spontaneous breakdown of a global symmetry provides the massless Goldstone modes that are necessary for dimensional reduction. On the other hand, when a gauge symmetry breaks spontaneously, the Higgs mechanism gives mass to the gauge bosons and dimensional reduction would not occur. Fortunately, non-Abelian gauge theories in five dimensions are generically in a massless Coulomb phase [26]. This has been verified in detail for 5-d SU (2) and SU (3) lattice gauge theories using Wilson’s formulation [6]. For sufficiently large representations of the embedding algebra SU (2N ) the same is true for quantum link models [7]. Whether a (4 + 1)-d SU (N ) quantum link model with a small representation of SU (2N ) is still in the Coulomb phase can only be checked in numerical simulations. The leading terms in the low-energy effective action of 5-d Coulombic gluons take the form   β 1 dx5 d4 x 2 (Tr Fµν Fµν S[A] = 2e 0 1 + Tr ∂5 Aµ ∂5 Aµ ). (23) c2 The quantum link model leads to a 5-d gauge theory characterized by the “velocity of light” c. Note that here µ runs over 4-d indices only. The dimensionful 5-d gauge coupling is given by 1/e2 . At finite β the above theory has only a 4-d gauge invariance, because we have fixed A5 = 0, i.e. we have not imposed the Gauss law. At β = ∞ we are in the 5-d Coulomb phase with massless gluons and thus with an infinite correlation length ξ. When β is made finite, the extent of the extra dimension is negligible compared to ξ. Hence, the theory appears to be dimensionally reduced to four dimensions. Of course, in four dimensions the confinement hypothesis suggests that gluons are no longer massless. Indeed, as it was argued in [2], a finite correlation length
ξ ∝ βc

11e2 N 48π 2 β




51/121 exp

24π 2 β 11N e2

 (24)

is expected to be generated nonperturbatively. Here 11N/48π 2 is the 1-loop β-function coefficient of SU (N ) gauge theory. For large β the gauge coupling of the dimensionally reduced 4-d

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theory is given by 1 β = 2. g2 e

(25)

Thus the continuum limit g → 0 of the 4-d theory is approached when one sends the extent β of the fifth direction to infinity. Hence, dimensional reduction occurs when the extent of the fifth direction becomes large. This is due to asymptotic freedom which implies that the correlation length grows exponentially with β. It is useful to think of the dimensionally reduced 4-d theory as a Wilsonian lattice theory with lattice spacing βc (which has nothing to do with the lattice spacing of the quantum link model). In fact, one can imagine performing a block renormalization group transformation that averages the 5-d field over cubic blocks of size β in the fifth direction and of size βc in the four physical space-time directions. The block centers then form a 4-d space-time lattice of spacing βc and the effective theory of the block averaged 5-d field is indeed a Wilsonian 4-d lattice gauge theory. 6. Quantum Link Models with Quarks To represent full QCD, it is essential to formulate quantum link models with quarks. This is more or less straightforward, although some subtleties arise related to the dimensional reduction of fermions. Before we discuss the quantum link formulation of full QCD, let us review the standard formulation of lattice gauge theory with fermions. The standard Wilson action with quarks is given by  ¯ ψ, u] = − Tr[ux,µ ux+ˆµ,ν u† x+ˆν ,µ u† x,ν ] S[ψ, x,µ=ν

1 ¯ [ψx γµ ux,µ ψx+ˆµ − ψ¯x+ˆµ γµ u† x,µ ψx ] 2 x,µ r ¯ + [2ψx ψx − ψ¯x ux,µ ψx+ˆµ − ψ¯x+ˆµ u† x,µ ψx 2 x,µ  +M (26) ψ¯x ψx ]. +

x

Here ψ¯x and ψx are independent Grassmann valued spinors associated with the lattice site x, γµ are Dirac matrices, and M is the bare quark mass.

The term proportional to r is the Wilson term that removes unwanted lattice fermion doublers at the expense of explicitly breaking chiral symmetry. In order to reach the continuum limit the bare mass M must be tuned appropriately. The guiding principle in the formulation of quantum link models is to replace the classical action of the standard formulation by an action operator that describes the evolution of the system in a fifth Euclidean direction. For the quarks we must also replace ψ¯x by Ψ† x γ5 . Hence, in the D-theory formulation the full QCD quantum link action operator is given by  Tr[Ux,µ Ux+ˆµ,ν U † x+ˆν ,µ U † x,ν ] H=J x,µ=ν

+J





[detUx,µ + detU † x,µ ]

x,µ

1 † + [Ψ x γ5 γµ Ux,µ Ψx+ˆµ 2 x,µ −Ψ† x+ˆµ γ5 γµ U † x,µ Ψx ] r + [2Ψ† x γ5 Ψx − Ψ† x γ5 Ux,µ Ψx+ˆµ 2 x,µ  −Ψ† x+ˆµ γ5 U † x,µ Ψx ] + M Ψ† x γ5 Ψx .

(27)

x

Here Ψ† x and Ψx are quark creation and annihilation operators obeying canonical anticommutation relations. Of course, we have again replaced the classical link variables ux,µ by quantum link operators Ux,µ . The generator of an SU (N ) gauge transformation now takes the form  x =  x−ˆµ,µ + L  x,µ ) + Ψ† xλΨx , G (R (28) µ

and it is again straightforward to show that H  x for all x. commutes with G The same universality arguments that were used before now suggest that the effective action of the corresponding 5-d gauge theory (with A5 = 0) takes the form  β  1 ¯ ψ, Aµ ] = S[ψ, dx5 d4 x { 2 [TrFµν Fµν 2e 0 1 + 2 Tr∂5 Aµ ∂5 Aµ ] c ¯ µ (Aµ + ∂µ ) + M + 1 γ5 ∂5 ]ψ}. +ψ[γ (29) c

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The “velocity of light” c , which characterizes the propagation of a quark in the fifth direction, is in general different from the corresponding quantity c for the gluons, because the quantum link formulation has no symmetry between the four physical space-time directions and the extra fifth direction. This is no problem, because we are only interested in the 4-d physics after dimensional reduction. To ensure the proper dimensional reduction of the quarks, their boundary conditions in the fifth direction must be chosen appropriately. The standard antiperiodic boundary conditions, which are dictated by thermodynamics in the Euclidean time direction, would lead to Matsubara modes, p5 = 2π(n5 + 21 )/β, which would limit the physical correlation length of the dimensionally reduced fermion to O(βc ). The confinement physics of the induced 4-d gluon theory, on the other hand, takes place at a correlation length which is growing exponentially with β. In fact, βc plays the role of the lattice spacing of the dimensionally reduced theory. Quarks with antiperiodic boundary conditions in the fifth direction would hence remain at the cut-off and the dimensionally reduced theory would still be a Yang-Mills theory without quarks. Once this problem is understood, one potential solution seems obvious. One may simply choose periodic boundary conditions for the quarks in the fifth direction. This gives rise to a Matsubara mode, p5 = 0, that survives dimensional reduction. Since the extent of the fifth direction has nothing to do with the inverse temperature (which is the extent of the Euclidean time direction), one could indeed choose the boundary condition in this way. However, the above scenario with periodic boundary conditions for the quarks would suffer from the same fine-tuning problem as the original Wilson fermion method. The bare quark mass would have to be adjusted very carefully in order to reach the chiral limit. This problem has been solved very elegantly in Shamir’s variant [27] of Kaplan’s fermion proposal [17]. Kaplan studied the physics of a 5-d system of fermions, which is always vector-like, coupled to a 4-d domain wall that manifests itself as a topological defect. The key observation is that under these conditions a zero mode of the

5-d Dirac operator appears as a bound state localized on the domain wall. From the point of view of the 4-d domain wall, the zero mode represents a massless chiral fermion. The original idea was to construct lattice chiral gauge theories in this way. Shamir has pointed out that the same mechanism can solve the lattice fine-tuning problem of the bare fermion mass in vector-like theories including QCD. He also suggested a variant of Kaplan’s method that has several technical advantages and that turns out to fit very naturally with the construction of quantum link QCD. In quantum link models we already have a fifth direction for reasons totally unrelated to the chiral symmetry of fermions. We will now use the fifth direction to solve the fine-tuning problem that we would have with periodic boundary conditions for the quarks. Shamir’s technical simplification compared to Kaplan’s original proposal is that one now works with a 5-d slab of finite size β with open boundary conditions for the fermions at the two sides. This geometry limits one to vector-like theories, because now there are two zero modes — one at each boundary — which correspond to one leftand one right-handed fermion in four dimensions. This set-up fits naturally with our construction of quantum link QCD. In particular, the evolution of the system in the fifth direction is still governed by the action operator of eq.(27). The only (but important) difference to Wilson’s fermion method is that now r < 0. Of course, one could also obtain a left- and a right-handed fermion by using a domain wall and an anti-wall with otherwise periodic boundary conditions. In that case the action operator of eq.(27) would have to be modified in an x5 -dependent way. Shamir’s method is more economical and concentrates on the essential topological aspects, which are encoded in the boundary conditions for the fermions in the fifth direction. It is important that in Shamir’s construction one also puts A5 = 0. There are some differences between the implementations of the method in the standard formulation of lattice gauge theory and in quantum link models. In the standard formulation one works with a 4-d gauge field, which is constant in the fifth direction. In quantum link QCD this is not possible,

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because the nontrivial dynamics in the fifth direction turns the discrete states of quantum links into the continuous degrees of freedom of physical gluons. However, it is still true that the physical gluon field is essentially constant in the fifth direction, because its correlation length grows exponentially with β. This is important for the generation of the fermionic zero modes at the two sides of the 5-d slab. The confinement physics of quantum link QCD in the chiral limit takes place at a length scale 24π 2 β 1 ∝ exp( ), m (11N − 2Nf )e2

(30)

which is determined by the 1-loop coefficient of the β-function of QCD with Nf massless quarks and by the 5-d gauge coupling e. As long as one chooses M>

24π 2 , (11N − 2Nf )e2

(31)

the chiral limit is reached automatically when one approaches the continuum limit by making β large. For a given value of r one is limited by M < −2r (note that r < 0). On the other hand, one can always choose J (and thus e2 ) such that the above inequality is satisfied. 7. Conclusions We have seen that D-theory provides a rich algebraic structure which allows us to formulate quantum field theories in terms of discrete variables — quantum spins or quantum links. Dimensional reduction of discrete variables is a generic phenomenon. In (d + 1)-dimensional quantum spin models with d ≥ 2, it occurs because of spontaneous symmetry breaking, while in (4 + 1)dimensional non-Abelian quantum link models it is due to the presence of a 5-d massless Coulomb phase. The inclusion of fermions is very natural when one follows Shamir’s variant [27] of Kaplan’s domain wall fermion proposal [17]. In particular, the fine-tuning problem of Wilson fermions is solved very elegantly by going to five dimensions. It is remarkable that D-theory treats bosons and fermions on an equal footing. Both are formulated in a finite Hilbert space per site, both

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require the presence of an extra dimension, and both naturally have exponentially large correlation lengths after dimensional reduction. The discrete nature of the fundamental variables makes D-theory attractive, both from an analytic and from a computational point of view. On the analytic side, the discrete variables allow us to rewrite the theory in terms of fermionic constituents of the bosonic fields. This may turn out to be useful when one studies the large N limit of various models [8]. In particular, one can now carry over powerful techniques developed for condensed matter systems (like the quantum Heisenberg model) to particle physics. This includes the use of very efficient cluster algorithms which has the potential of dramatically improving numerical simulations of lattice field theories. In D-theory the classical fields of ordinary quantum field theory arise via dimensional reduction of discrete variables. This requires specific dynamics — namely a massless theory in one more dimension. In general, the verification of this basic dynamical ingredient of D-theory requires nonperturbative insight — for example, via numerical simulations or via the large N limit. Thus, the connection to ordinary field theory methods — in particular, to perturbation theory — is somewhat indirect. This could be viewed as a potential weakness, for example, because it seems hopeless to do perturbative QCD calculations in the framework of D-theory. However, the large separation from perturbative methods may actually turn out to be a major strength of Dtheory. The fact, that perturbative calculations are difficult, may imply that nonperturbative calculations are now easier. After all, D-theory provides an additional nonperturbative microscopic structure underlying Wilson’s lattice theory. One may hope that this structure will help us to better understand the nonperturbative dynamics of quantum field theories. Besides potential conceptual and numerical advantages for solving nonperturbative problems in field theory, in this talk we also contemplate that D-theory may resemble Nature’s physical regularization. Although this is highly speculative, we like to point out that D-theory indeed offers room for nonperturbative thought on fundamen-

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tal physics alternative to string theory. Of course, the present constructions with a rigid lattice and just one extra dimension may not be sufficient, but the idea that the most fundamental degrees of freedom are discrete quantum variables may lead to fruitful developments. Of course, it should be admitted right away that the full standard model (or even just an Abelian chiral gauge theory) has not yet been regularized within the D-theory framework. Also gravity has not yet emerged from some underlying quantum spin dynamics. These represent great challenges that are worth facing. However, one should not forget that, beyond perturbation theory, string theory has not yet solved these hard problems either. It goes without saying that any attempt to make statements about very high energy scales must necessarily remain highly speculative. After all, the rest of physics can be mastered only because low-energy effective theories (perturbatively renormalizable or not) are to a large extent insensitive to the details of high-energy physics. Of course, the particular values of low-energy parameters can only be understood in terms of more fundamental physics. For example, the parameters of the standard model, such as the values of the gauge couplings, the quark and lepton masses and mixing angles, as well as the number of colors, and the vacuum angle θ, depend on the details of Nature’s regularization at ultrashort distances. Still, inferring those details from the experimentally determined values of the standard model parameters is extremely difficult, if not impossible. If Nature at the Planck scale was indeed regularized by a D-theory quantum spin system, the situation would be similar to condensed matter physics. For example, just knowing the spin wave velocity and the spin stiffness, and perhaps a few more low-energy parameters of a magnet, it is virtually impossible to infer the fundamental QED dynamics underlying condensed matter physics. For the same reason, we will obviously have to wait for new experiments in order to gain deeper insight into the physics at ultrashort distances. If there are strings and branes, quantum spins and quantum links, or tiny wheels turning around at the Planck scale is hence likely to remain an open question in the foreseeable future.

Acknowledgements I like to thank B. B. Beard, R. Brower, S. Chandrasekharan, M. Pepe, and S. Riederer for a very pleasant collaboration on the subjects discussed in this talk. I’m also indebted to J. Goldstone for a remark about the hierarchy problem he made a long time ago. This work is supported in parts by the Schweizerischer Nationalfonds (SNF). REFERENCES 1. P. A. M. Dirac, “The Inadequacies of Quantum Field Theory”, printed in “Paul Adrien Maurice Dirac: Reminiscences about a great Physicist”, eds. B. N. Kursunoglu and E. P. Wigner, Cambridge University Press (1987). 2. S. Chandrasekharan and U.-J. Wiese, Nucl. Phys. B492 (1997) 455. 3. R. Brower, S. Chandrasekharan, and U.J. Wiese, Phys. Rev. D60 (1999) 094502. 4. U.-J. Wiese, Prog. Theor. Phys. Suppl. 131 (1998) 483. 5. U.-J. Wiese, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 146. 6. B. B. Beard, R. C. Brower, S. Chandrasekharan, D. Chen, A. Tsapalis, and U.-J. Wiese, Nucl. Phys. (Proc. Suppl.) 63 (1998) 775. 7. B. Schlittgen and U.-J. Wiese, Phys. Rev. D63 (2001) 085007. 8. O. B¨ ar, R. C. Brower, B. Schlittgen, and U.-J. Wiese, Nucl. Phys. (Proc. Suppl.) 106 (2002) 1019. 9. S. Chandrasekharan, B. Scarlet, and U.J. Wiese, Comput. Phys. Commun. 147 (2002) 388. 10. R. C. Brower, S. Chandrasekharan, S. Riederer, and U.-J. Wiese, Nucl. Phys. B693 (2004) 149. 11. B. B. Beard, M. Pepe, S. Riederer, and U.J. Wiese, Phys. Rev. Lett. 94 (2005) 010603. 12. F. Wilczek, hep-ph/0201222. 13. S. Chandrasekharan and U.-J. Wiese, Prog. Prat. Nucl. Phys. 53 (2004) 373. 14. K. Wilson, Phys. Rev. D10 (1974) 2445. 15. H. B. Nielsen and M. Ninomiya, Phys. Lett. B105 (1981) 219; Nucl. Phys. B185 (1981) 20.

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16. D. Friedan, Commun. Math. Phys. 85 (1982) 481. 17. D. B. Kaplan, Phys. Lett. B288 (1992) 342. 18. V. A. Rubakov and M. Shaposhnikov, Phys. Lett. B125 (1983) 136. 19. R. Narayanan and H. Neuberger, Phys. Lett. B302 (1993) 62; Phys. Rev. Lett. 71 (1993) 3251; Nucl. Phys. B412 (1994) 574. 20. M. L¨ uscher, Phys. Lett. B428 (1998) 342. 21. P. H. Ginsparg and K. G. Wilson, Phys. Rev. D25 (1982) 2649. 22. P. Hasenfratz, Nucl. Phys. Proc. Suppl. 63 (1998) 53. 23. D. B. Kaplan, Nucl. Phys. Proc. Suppl. 129 (2004) 109. 24. M. L¨ uscher, Nucl. Phys. B549 (1999) 295; Nucl. Phys. B568 (2000) 162. 25. O. B¨ ar and I. Campos, Nucl. Phys. B581 (2000) 499. 26. M. Creutz, Phys. Rev. Lett. 43 (1979) 553. 27. Y. Shamir, Nucl. Phys. B406 (1993) 90.

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