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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 236783, 15 pages
Evolving Center-Vortex Loops
Institut für Theoretische Physik, Universität Karlsruhe (TH), Kaiserstraße 12, 76131 Karlsruhe, Germany
Received 7 December 2011; Accepted 29 March 2012
Academic Editors: J. Bičák and M. Ehrnström
Copyright © 2012 Julian Moosmann and Ralf Hofmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider coarse-graining applied to nonselfintersecting planar centervortex loops as they emerge in the confining phase of an SU(2) Yang-Mills theory. Well-established properties of planar curve-shrinking predict that a suitably defined, geometric effective action exhibits (mean-field) critical behavior when the conformal limit of circular points is reached. This suggests the existence of an asymptotic mass gap. We demonstrate that the initially sharp mean center-of-mass position in a given ensemble of curves develops a variance under the flow as is the case for a position eigenstate in free-particle quantum mechanics under unitary time evolution. A possible application of these concepts is an approach to high- superconductivity based (a) on the nonlocal nature of the electron (1 fold selfintersecting center-vortex loop) and (b) on planar curve-shrinking flow representing the decrease in thermal noise in a cooling cuprate.
The problem of how a confining ground state emerges in 4D nonabelian Yang-Mills theory was addressed by ’t Hooft in terms of the definition of an order parameter that is dual to the Wilson loop . The lattice-gauge-theory measurement as well as analytical model approaches to the ’t Hooft loop expectation was performed by several authors starting shortly thereafter [2–7]. The according confinement scenario is a condensation of magnetic center vortices which induces narrow electric flux lines in between color-electric sources granting a potential which rises sufficiently rapid with an increasing distance of separation. Due to the absence of magnetic center monopoles the associated center vortices form closed flux loops.
Four-dimensional SU Yang-Mills theory occurs in three phases: a deconfining, a preconfining, and a confining one. While the former two phases possess propagating gauge fields, a complete decoupling thereof takes place at a Hagedorn transition towards the confining phase [8–12]. Namely, by the decay of a preconfining ground state, consisting of collapsing magnetic (w.r.t. the gauge fields in the defining SU Yang-Mills Lagrangian) flux lines of finite core-size , see also , into nonselfintersecting or selfintersecting center-vortex loops , the mass of the dual gauge field diverges. This, in turn, implies . As a consequence, center-vortex loops (CVLs) with nonvanishing selfintersection number become stable solitons in isolation. These solitons are classified according to their topology and center charge. That is, for , the region of negative pressure is confined to the vanishing vortex core, and the soliton becomes a particle-like () excitation whose stability is in addition assured topologically by its selfintersection(s).
The purpose of our paper is to investigate the sector with in some detail. Topologically, there is no reason for the stability of this sector’s excitations, and we will argue that on average, and as a consequence of a noisy environment a planar CVL with shrinks to nothingness within a finite “time.” Here the role of “time” is played by a variable measuring the decrease of externally provided resolving power applied to the system. By “planar CVL” we mean an embedding of the soliton into a 2D flat and spatial plane. For an isolated SU Yang-Mills theory the role of the environment is played by the sectors with . If the SU theory under consideration is part of a world subject to additional gauge symmetries, then a portion of such an environment arises from a mixing with these theories. In any case, a planar CVL at finite length is acquiring mass by frequent interactions with the environment after it was generated by a process subject to an inherent, finite resolution . At the same time, the CVL starts shrinking towards a circular point. The latter becomes unresolvable starting at some finite resolution . That is, all properties that are related to the existence of extended lines of center flux, observable for , do not occur for , and the CVL vanishes from the spectrum of confining SU Quantum Yang-Mills theory. Since CVLs with have a finite mass (positive-integer multiples of the Yang-Mills scale ), we “observe” a gap in the mass spectrum of the theory when probing the system with resolution .
The reader may object that the restriction of the shrinking dynamics to a 2D plane does not justify the conclusion about the aforementioned occurrence of a mass gap since CVLs evolve in 3D. However, according to the work of [14, 15] and for low-energy SU Yang-Mills theory the assumption of planar, closed, and smooth curves accurately describing the late-“time” behavior of the evolution of any given center-vortex loop with is valid because the CVL becomes planar after a finite “time” under the curve shrinking equation. Thus, to discuss the approach of the round-point limit (mass-gap property) we may pick planar initial conditions and consider the subsequent evolution in the plane. The formation of a singularity (a loop pinching off to form a cusp) is a planar phenomenon  which, however, is excluded to occur in low-energy, that is, weakly resolved SU Yang-Mills theory because the energy required to form and resolve such a singularity is not available.
Notice that by embedding an CVL of an isolated SU Yang-Mills theory into a flat 2D surface at , , a hypothetical observer measuring a positive (negative) curvature along a given sector of the vortex line experiences more (less) negative pressure in the intermediate vicinity of this curve sector leading to this sector’s motion towards (away from) the observer, see Figure 1. The speed of this motion is a monotonic function of curvature. On average, this shrinks the CVL. Alternatively, one may globally consider the limit , , that is, the confining phase of an SU Yang-Mills theory, but now take into account the effects of an environment which locally relaxes this limit (by collisions) and thus also induces curve-shrinking. One possibility to describe this situation is by the following flow equation in the (dimensionless) parameter where is arc length, is a point on the planar CVL, and is a string tension effectively expressing the distortions induced by the environment. After rescaling, , , (1.1) assumes the following form: where is a (dimensionless) curve parameter (), the (inward-pointing) Euclidean unit normal, and the scalar curvature, defined as, and denotes the Euclidean scalar product of the vectors and lying in the plane. In the following, we resort to a slight abuse of notation by using the same symbol for the functional dependence on or .
It is worth mentioning that (1.2) expresses a special case of the local condition that the rate of decrease of the (dimensionless) curve length is maximal w.r.t. a variation of the direction of the velocity of a given point on the curve at fixed : The 1D heat equation (1.2) is well understood mathematically [17, 18]. The evolution of a particular initial configuration of a planar CVL () into the round-point limit () is depicted in Figure 2.
The present work interprets the shrinking of closed and 2D flat-embedded (planar) curves as a Wilsonian renormalization group evolution governed by an effective action, which is defined purely in geometric terms. In the presence of an environment represented by the parameter , this action possesses a natural decomposition into a conformal and a nonconformal factor. One of our goals is to show that the transition to the conformal limit of vanishing mean curve length really is a critical phenomenon characterized by a mean-field exponent if a suitable parameterization of the effective action is used. To see this, various initial conditions are chosen to generate an ensemble whose partition function is invariant under curve shrinking. A (second-order) phase transition is characterized by the critical behavior of the coefficient associated with the nonconformal factor in the effective action. That is, in the presence of an environment the (nearly massless) sector in confining SU Yang-Mills dynamics, generated during the Hagedorn transition, practically disappears after a finite “time” leading to an asymptotic mass gap. We also believe that CVLs play the role of Majorana neutrinos in physics: their disappearance after a finite time and the absence of antiparticles would be manifestations of lepton-number violation [19–21] forbidden in the present standard model of particle physics, see  for a discussion of experimental signatures in this context.
In Section 2, we provide information on proven properties of curve-shrinking evolution. The philosophy underlying the statistics of geometric fluctuations in the sector is elucidated in Section 3. In Section 4, we present our results for the renormalization-group flow of the effective action, give an interpretation, and compute the evolution of a local quantity. Finally, in Section 5 we summarize our work and give an outlook to the case which we suspect to be relevant to high superconductivity.
2. Prerequisites on Mathematical Results
In this section, we provide knowledge on the properties of the shrinking of embedded (nonselfintersecting) curves in a plane [17, 18]. It is important to stress that only for curve-shrinking in a plane are the following results valid. To restrict the motion of a CVL to a plane, however, is justified when physics close to the round-point limit is considered [14, 15], see Section 1.
The properties of the -evolution of smooth, embedded, and closed curves subject to (1.2) were investigated in  for the purely convex case and in  for the general case. The main result of  is that an embedded curve with finitely many points of inflection remains embedded and smooth when evolving under (1.2) and that such a curve flows to a circular point for , where . That is, asymptotically the curve converges (w.r.t. the -norm) to a shrinking circle: and , being the (dimensionless) area enclosed by the curve, such that the isoperimetric ratio approaches the value from above. For later use, we present the following two identities, see Lemmas and in : Setting , the solution to (2.2) is By virtue of (2.3) the critical value is related to as
3. Geometric Partition Function
We now wish to interpret curve-shrinking as a Wilsonian renormalization-group flow taking place in the planar CVL sector. A partition function, defined as a statistical average (according to a suitably defined weight) over CVLs, is to be left invariant under a decrease of the resolution determined by the flow parameter . Notice that, physically, is interpreted as a strictly monotonic decreasing (dimensionless) function of a ratio where () are mass scales associated with an actual (initial) resolution applied to the system.
To devise a geometric ansatz for the effective action , which is a functional of the curve representable in terms of integrals over local densities in (reparametrization invariance), the following reflection on symmetries is in order. (i) scaling symmetry : For both, and , implying and at fixed , the action should be invariant under further finite rescalings (decoupling of the fixed length scale ), (ii) Euclidean point symmetry of the plane (rotations, translations and reflections about a given axis): Sufficient but not necessary for this is a representation of in terms of integrals over scalar densities w.r.t. these symmetries. That is, the action density should be expressible as a series involving products of Euclidean scalar products of , , or constancy. However, an exceptional scalar integral over a nonscalar density can be devised. Consider the area , calculated as The density in (3.1) is not a scalar under translations.
We now resort to a factorization ansatz as where in addition to Euclidean point symmetry () is (is not) invariant under . In principle, infinitely many operators can be defined to contribute to . Since the evolution generates circles for higher derivatives of w.r.t. rapidly converge to zero . We expect this to be true also for Euclidean scalar products involving higher derivatives . To yield conformally invariant expressions, such integrals need to be multiplied by powers of and/or or the inverse of integrals involving lower derivatives. At this stage, we are not capable of constraining the expansion in derivatives by additional physical or mathematical arguments. To be pragmatic, we simply set equal to the isoperimetric ratio: We conceive the nonconformal factor in as a formal expansion in inverse powers of . Since we regard the renormalization-group evolution of the effective action as induced by the flow of an ensemble of curves, where the evolution of each member is dictated by (1.2), we allow for an explicit dependence of the coefficient of the lowest nontrivial power . In principle, this sums up the contribution to of certain higher-power operators which do not exhibit an explicit dependence. Hence, we make the following ansatz The initial value is determined from a physical boundary condition such as the mean length at which determines the mean mass of a CVL as .
For latter use, we investigate the behavior of as for an ensemble consisting of a single curve only and require the independence of the “partition function” under changes in . Integrating (2.1) in the vicinity of under the boundary condition that , we have Since independence of the “partition function” under the flow in implies that That is, approaches constancy for which brings us back to the conformal limit by a finite renormalization of the conformal part of the action. In this parameterization of , can thus be regarded as an order parameter for conformal symmetry with mean-field critical exponent.
4. Renormalization-Group Flow
4.1. Effective Action
Let us now numerically investigate the effective action resulting from a partition function w.r.t. a nontrivial ensemble . The latter is defined as the average over the ensemble . Let us denote by an ensemble consisting of curves where is obtained from by adding a new curve . In Figure 3, eight initial curves are depicted which in this way generate the ensembles for .
We are interested in a situation where all curves in shrink to a point at the same value . Because of (2.3) and (2.4), we thus demand that at all curves in initially have the same area . The effective action in (3.2) (when associated with the ensemble we will denote it as ) is determined by the function , compared with (3.4), whose flow follows from the requirement of -independence of : This is an implicit, first-order ordinary differential equation for which needs to be supplemented with an initial condition . A natural initial condition is to demand that the quantity: coincides with the arithmetic mean defined as From a value for follows. We also have considered a modified factor in (3.2). In this case the choice of initial condition leads to . While the geometric effective action thus is profoundly different for such a modification of physical results such as the evolution of the variance of center-of-mass position agree remarkably well, see Section 4.2. That is, the geometric effective action itself is not a physical object. Rather, going from one ansatz for to another describes a particular way of redistributing the weight in the ensemble which seems to have no significant impact on the physics. This is in contrast to quantum field theory and conventional statistical mechanics, where the action in principle is related to the physical properties of a given member of the ensemble.
For the curves depicted in Figure 3, we make the convention that . It then follows that by (2.4). The dependence is plotted in Figure 4. According to Figure 4, it seems that the larger the ensemble the closer to the evolution of a single circle of initial radius . That is, for growing the function approaches the form: where the slope depends on the strength of deviation from circles of the representatives in the ensemble at , that is, on the variance at a given value . Physically speaking, the value is associated with a certain initial resolution of the measuring devise (the strictly monotonic function , being a physical scale such as energy or momentum transfer, expresses the characteristics of the measuring device and the measuring process), the value of describes the strength of noise associated with the environment ( determines how fast the conformal limit of circular points is reached), and the values of and , see (4.5), are associated with the conditions at which the to-be-coarse-grained system is prepared. Notice that this interpretation is valid for the action only.
4.2. Statistical Uncertainty of Center of Mass Position
We are now in a position to compute the flow of a more local “observable,” namely, the mean “center-of-mass” (COM) position in a given ensemble and the statistical variance of the COM position. The COM position of a given curve is defined as We will below present only results on the statistical variance of the COM position.
Let as assume that at , the ensembles are modified such that a translation is applied to each representative letting its COM position coincide with the origin. Recall that such a modification does not alter the (effective) action (Euclidean point symmetry). That is, at the statistical variance in the position of the COM is prepared to be nil, physically corresponding to an infinite resolution applied to the system by the measuring device.
The mean COM position over ensemble is defined as The scalar statistical deviation of over the ensemble is defined as where and similarly for the coordinate . In Figure 5, plots of are shown when is evaluated over the ensembles with the action: and subject to the initial condition . In Figure 5(b) the according plots of are depicted as obtained with the action and subject to the initial condition . In this case, one has leading to equal weights for each curve in .
4.3. Quantum Mechanical versus Statistical Uncertainty
In view of the results obtained in Section 4.2, we would say that an ensemble of evolving planar CVLs in the sector qualitatively resembles the quantum mechanics of a free-point particle (it is of no relevance at this point whether this particle carries spin or not) of mass in 1D. Namely, an initially localized square of the wave function with , where , according to unitary time evolution in quantum mechanics evolves as , where is the free-particle Hamiltonian, is the spatial momentum, and . In agreement with Heisenberg’s uncertainty relation one has during the process that . Time evolution in quantum mechanics and the process of coarse-graining in a statistical system describing planar CVLs share the property that in both systems the evolution generates out of a small initial position uncertainty (corresponding to a large initial resolution ) a larger position uncertainty in the course of the evolution. Possibly, future development will show that interference effects in quantum mechanics can be traced back to the nonlocal nature of the degrees of freedom (CVLs) entering a statistical partition function.
5. Summary, Conclusions, and Outlook
5.1. Summary of Present Work
In this exploratory paper an attempt has been undertaken to interpret the effects of an environment on 2D planar center-vortex loops, as they emerge in the confining phase of an SU Yang-Mills theory, in terms of a Wilsonian renormalization-group flow carried by purely geometric entities. Our (mainly numerical) analysis uses established mathematics on the shrinking of embedded curves in the plane. In the case of nonintersecting CVLs () the role of the environment is played by the entirety of all sectors with and, possibly, an explicit environment. In a particular parametrization of the effective action we observe critical behavior as the limit of circular points is approached during the evolution. That is, planar CVLs on average disappear from the spectrum for resolving powers smaller than a critical, finite value. Using this formalism to compute the evolution of the mean values of local observables, such as the center-of-mass position, a behavior is generated that qualitatively resembles the associated unitary evolution in quantum mechanics. We also have found evidence that this situation is practically not altered when changing the ansatz for the effective action.
5.2. Outlook on Strongly Correlated Electrons in a Plane
Let us conclude this article with a somewhat speculative outlook on planar CVLs. Setting the Yang-Mills scale (mass of the intersection in the CVL) of the associated SU theory equal to the electron mass, this soliton is interpreted as an electron or positron, see [8, 9, 23, 24] for a more detailed discussion on the viability of such an assignment and Figure 6 for a display. Important for our purpose are the facts that the two-fold directional degeneracy of the center flux represents the two-fold degeneracy of the spin projection and that a large class of curve deformations (shifts of the intersection point) leaves the mass of the soliton invariant in the free case.
It is a remarkable fact that a high level of mathematical understanding exists for the behavior of curve-shrinking in a 2D plane [17, 18] even in the case of one selfintersection . Incidentally, 2D quantum systems exist in nature which exhibit unconventional behavior. Specifically, the phenomenon of high superconductivity appears to be strongly related to the two-dimensionality of electron dynamics as it is enabled by rare-earth doping of cuprate materials . Apparently, the Coulomb repulsion between the electrons moving in the would-be valence band within the cuprate planes of high superconductors (Mott insulators) is effectively screened by the interplane environment also providing for the very existence of these electrons by doping. The question then is how the long-range order of electronic spins, which at given (optimal) doping and at a sufficiently low temperature leads to superconductivity, emerges within the cuprate planes. As it seems, quantum Monte Carlo simulations of a transformed Hubbard model (- model) subject to Gutzwiller projection yield quantitative explanations of a number of experimental results related to the existence of the pseudogap phase (Nernst effect, nonlinear diamagnetic susceptibility), see [27, 28] and references therein.
We would here like to offer a sketch of an alternative approach to high superconductivity being well aware of our ignorance on the details of present-day research in this field. The key idea is already encoded in Figure 6. Namely, according to confining SU Yang-Mills theory the electron is a nonlocal object with the physics of its charge localization being only loosely related to the physics of its magnetic moment (spin): the magnetic moment, carried by the core of the flux line, microscopically manifested by (oppositely) moving (opposite) electric charges, receives contributions from vortex sectors that are spatially far separated (on the scale of the diameter of the intersection point) from the location of the isolated electric charge. This suggests that in certain physical circumstances, where the ordering effect of interacting vortex lines becomes important, the postulate of a spinning point particle fails to describe reality.
Concerning the strong correlations in 2D electron dynamics responsible for high superconductivity, we imagine a situation as depicted in Figure 7. Each electron’s spin in the plane interacts with the spin of its neighbors as follows. Equally directed electric fluxes (dually interpreted center fluxes of ) attract one another, and there is attraction for four out of six vortex sectors defined by the neighboring electrons (Ampere’s law), while two vortex sectors experience repulsion (that the electric charges of confining are seen by the photon is a consequence of the mixing between the corresponding two gauge groups: and [8, 9, 24]). Notice that for a given electron two of the adjacent electrons exhibit equal spin projection, while four of the adjacent electrons have opposite spins. This is in agreement with the observation that high superconductivity is an effect not related to -wave Cooper-pair condensation.
An overlap of vortex sectors, hypothetically leading to the creation of extra intersection points, is topologically forbidden. That is, the fluctuations in the energy density of the system are far too weak to create an intersection point of mass keV. Therefore, at very small spatial separation repulsion must occur between adjacent vortex sectors. At a sufficiently low temperature and an optimal screening of Coulomb repulsion by the interplane environment (doping), this would lead to a typical equilibrium configuration as depicted in Figure 7 where the intersection points (electronic charge) do not move relative to one another. A local demolition of this highly ordered state would cost a finite amount of energy manifested in terms of the (gigantic) gaps measured experimentally in the cuprate systems. Applying an electric field vector with a component parallel to the plane would set into resistivity-free motion the thus locked electrons. For a macroscopic analogue, imagine a stiff table cloth being pulled over the table’s surface in a friction-free fashion. The occurrence of the pseudogap phase would possibly be explained by local defects in the fabric of Figure 7 due to insufficient Coulomb screening and/or too large thermal noise (macroscopic vorticity; liquid of point-like defects in 2D).
The authors would like to thank Markus Schwarz for useful conversations.
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