La cuantificación del frente de luz [1] [2] [3] de las teorías cuánticas de campo proporciona una alternativa útil a la cuantificación ordinaria en tiempos iguales . En particular, puede conducir a una descripción relativista de los sistemas ligados en términos de funciones de onda de la mecánica cuántica . La cuantificación se basa en la elección de las coordenadas del frente de luz, [4] donde juega el papel del tiempo y la correspondiente coordenada espacial es . Aquí, es el tiempo ordinario, es una coordenada cartesiana , yes la velocidad de la luz. Las otras dos coordenadas cartesianas, y , están intactos y a menudo se les llama transversales o perpendiculares, denotados por símbolos del tipo . La elección del marco de referencia donde el tiempo y Los ejes definidos pueden dejarse sin especificar en una teoría relativista exactamente soluble, pero en cálculos prácticos algunas opciones pueden ser más adecuadas que otras.
La solución de la ecuación de valor propio hamiltoniano LFQCD utilizará los métodos matemáticos disponibles de la mecánica cuántica y contribuirá al desarrollo de técnicas informáticas avanzadas para grandes sistemas cuánticos, incluidos los núcleos . Por ejemplo, en el método de cuantificación de cono de luz discretizado (DLCQ), [5] [6] [7] [8] [9] [10] se introducen condiciones periódicas tales que los momentos se discretizan y el tamaño del espacio de Fock es limitado sin destruir la invariancia de Lorentz. La resolución de una teoría cuántica de campos se reduce entonces a diagonalizar una gran matriz hermitiana dispersa . El método DLCQ se ha utilizado con éxito para obtener el espectro completo y las funciones de onda de frente de luz en numerosas teorías de campo cuántico modelo, como QCD con una o dos dimensiones espaciales para cualquier número de sabores y masas de quarks. Una extensión de este método a las teorías supersimétricas , SDLCQ, [11] [12] aprovecha el hecho de que el hamiltoniano de frente ligero se puede factorizar como un producto de operadores de escalera de subida y bajada . SDLCQ ha proporcionado nuevos conocimientos sobre una serie de teorías supersimétricas, incluida la evidencia numérica directa [13] de una dualidad supergravedad / super-Yang-Mills conjeturada por Maldacena.
Es conveniente trabajar en base a Fock. donde los momentos del frente de luz y son diagonales. El estado está dado por una expansión
con
se interpreta como la función de onda de la contribución de estados con partículas. El problema de los valores propioses un conjunto de ecuaciones integrales acopladas para estas funciones de onda. Aunque la notación tal como se presenta solo admite un tipo de partícula, la generalización a más de una es trivial.
Cuantización discreta de cono de luz
Un enfoque sistemático para la discretización del problema de los valores propios es el método DLCQ sugerido originalmente por Pauli y Brodsky. [5] [6] En esencia, es el reemplazo de integrales por aproximaciones trapezoidales, con intervalos igualmente espaciados en los momentos longitudinal y transversal.
correspondiente a las condiciones de contorno periódicas en los intervalos y . Las escalas de longitud y determinar la resolución del cálculo. Debido a que el componente positivo del impulso es siempre positivo, el límitese puede cambiar por un límite en términos de resolución entera . La combinación de componentes de impulso que define es entonces independiente de . Las fracciones de impulso longitudinal convertirse en proporciones de números enteros . Porque el son todos positivos, DLCQ limita automáticamente el número de partículas a no más de . Cuando se proporciona un límite en el momento transversal a través de un punto de corte elegido, se obtiene un problema de matriz finita; sin embargo, la matriz puede ser demasiado grande para las técnicas numéricas actuales. Entonces se puede hacer un truncamiento explícito en el número de partículas, el equivalente de cono de luz de la aproximación de Tamm-Dancoff. Los tamaños de base grandes requieren técnicas especiales para la diagonalización de la matriz; el que se utiliza habitualmente es el algoritmo de Lanczos . Para el caso de una dimensión espacial, se puede resolver fácilmente el espectro de hadrones de QCD para cualquier masa y color de quark.
La mayoría de los cálculos de DLCQ se realizan sin modos cero. Sin embargo, en principio, cualquier base DLCQ con condiciones de contorno periódicas puede incluirlas como modos restringidos, dependiendo de los otros modos con impulso distinto de cero. La restricción proviene del promedio espacial de la ecuación de Euler-Lagrange para el campo. Esta ecuación de restricción puede ser difícil de resolver, incluso para las teorías más simples. Sin embargo, se puede encontrar una solución aproximada, consistente con las aproximaciones subyacentes del propio método DLCQ. [14] Esta solución genera las interacciones efectivas en modo cero para el hamiltoniano de frente de luz.
Los cálculos en el sector masivo que se realizan sin modos cero generalmente arrojarán la respuesta correcta. El descuido de los modos cero simplemente empeora la convergencia. Una excepción es la de las teorías escalares cúbicas, donde el espectro se extiende hasta menos infinito. Un cálculo DLCQ sin modos cero requerirá una extrapolación cuidadosa para detectar este infinito, mientras que un cálculo que incluye modos cero produce el resultado correcto de inmediato. Los modos cero se evitan si se utilizan condiciones de contorno antiperiódicas.
Cuantización de cono de luz discreta supersimétrica
La forma supersimétrica de DLCQ (SDLCQ) [11] [12] está diseñada específicamente para mantener la supersimetría en la aproximación discreta. El DLCQ ordinario viola la supersimetría por términos que no sobreviven al límite del continuo. La construcción SDLCQ discretiza la supercargay define el hamiltoniano por la relación superalgebra . El rango de la cantidad de movimiento transversal está limitado por un simple corte en el valor de la cantidad de movimiento. Se espera que los efectos de los modos cero se cancelen.
Además de los cálculos de espectros, esta técnica se puede utilizar para calcular los valores esperados. Una de esas cantidades, un correlacionador del tensor de energía de tensión , se ha calculado como una prueba de una conjetura de Maldacena . Para este cálculo se desarrolló un método basado en Lanczos muy eficiente. Los resultados más recientes proporcionan evidencia directa de la conjetura. [13]
Celosía transversal
El método de celosía transversal [15] [16] reúne dos ideas poderosas en la teoría cuántica de campos: la cuantificación hamiltoniana de frente de luz y la teoría del calibre de celosía. La teoría del calibre de celosía es un medio muy popular de regular para el cálculo las teorías de calibre que describen toda la materia visible en el universo; en particular, demuestra manifiestamente el confinamiento lineal de QCD que contiene quarks y gluones dentro de los protones y neutrones del núcleo atómico. En general, para obtener soluciones de una teoría cuántica de campos, con sus grados de libertad continuamente infinitos, se deben poner cortes cinemáticos u otras restricciones en el espacio de los estados cuánticos. Para eliminar los errores que esto introduce, se pueden extrapolar estos límites, siempre que exista un límite continuo, y / o volver a normalizar los observables para tener en cuenta los grados de libertad por encima del límite. Para los propósitos de la cuantificación hamiltoniana, se debe tener una dirección de tiempo continua. En el caso de la cuantificación hamiltoniana de frente de luz, además del tiempo de frente de luz continuo, es necesario mantener el dirección continua si se quiere preservar la invariancia de refuerzo de Lorentz manifiesta en una dirección e incluir pequeñas energías de frente de luz . Por lo tanto, como máximo se puede imponer un corte de celosía en las restantes direcciones espaciales transversales. Bardeen y Pearson sugirieron por primera vez esta teoría de la galga de celosía transversal en 1976. [15]
La mayoría de los cálculos prácticos realizados con la teoría del calibre de celosía transversal han utilizado un ingrediente adicional: la expansión dieléctrica de color. Una formulación dieléctrica es aquella en la que los elementos del grupo gauge, cuyos generadores son los campos de gluones en el caso de QCD, son reemplazados por variables colectivas (manchadas, bloqueadas, etc.) que representan un promedio sobre sus fluctuaciones en escalas de corta distancia. Estas variables dieléctricas son masivas, llevan color y forman una teoría de campo de calibre efectiva con la acción clásica minimizada en el campo cero, lo que significa que el flujo de color se expulsa del vacío al nivel clásico. Esto mantiene la trivialidad de la estructura de vacío de frente de luz, pero surge solo para un corte de momento bajo en la teoría efectiva (correspondiente a espaciamientos de celosía transversales de orden 1/2 fm en QCD). Como resultado, el hamiltoniano de corte efectivo inicialmente está pobremente restringido. No obstante, la expansión color-dieléctrica, junto con los requisitos de restauración de la simetría de Lorentz, se ha utilizado con éxito para organizar las interacciones en el hamiltoniano de una manera adecuada para una solución práctica. El espectro más preciso de grandes Se han obtenido bolas de pegamento de esta manera, así como funciones de onda de frente de luz de piones de acuerdo con una variedad de datos experimentales.
Cuantización básica de frente de luz
El enfoque de cuantización de base de frente de luz (BLFQ) [17] utiliza expansiones en productos de funciones de base de una sola partícula para representar las funciones de onda de estado de Fock. Normalmente, el longitudinal () la dependencia se representa en la base DLCQ de ondas planas , y la dependencia transversal se representa mediante funciones de oscilador armónico bidimensional . Estos últimos son ideales para aplicaciones de confinamiento de cavidades y son consistentes con QCD holográfico de frente de luz . [18] [19] [20] [21] [22] El uso de productos de funciones de base de una sola partícula también es conveniente para la incorporación de estadísticas de bosones y fermiones , porque los productos son fácilmente (anti) simetrizados. Al emplear funciones de base bidimensionales con simetría rotacional alrededor de la dirección longitudinal (donde las funciones del oscilador armónico sirven como ejemplo), se conserva el número cuántico de proyección del momento angular total que facilita la determinación del momento angular total de los estados propios de la masa. Para aplicaciones sin una cavidad externa, donde se conserva el momento transversal, se utiliza un método multiplicador de Lagrange para separar el movimiento transversal relativo del movimiento total del sistema.
La primera aplicación de BLFQ a QED resolvió el electrón en una cavidad de confinamiento transversal bidimensional y mostró cómo el momento magnético anómalo se comportaba en función de la fuerza de la cavidad. [23] La segunda aplicación de BLFQ a QED resolvió el momento magnético anómalo del electrón en el espacio libre [24] [25] y demostró concordancia con el momento de Schwinger en el límite apropiado.
La extensión de BLFQ al régimen dependiente del tiempo, es decir, BLFQ dependiente del tiempo (tBLFQ) es sencilla y actualmente se encuentra en desarrollo activo. El objetivo de tBLFQ es resolver la teoría del campo de frente de luz en tiempo real (con o sin campos de fondo dependientes del tiempo). Las áreas de aplicación típicas incluyen láseres intensos (consulte Cuantificación de frente de luz # Láseres intensos }) y colisiones relativistas de iones pesados .
Método de grupo acoplado de frente de luz
El método de clúster acoplado de frente de luz (LFCC) [26] es una forma particular de truncamiento para el sistema de ecuaciones integrales de acoplamiento infinito para funciones de onda de frente de luz. El sistema de ecuaciones que proviene de la ecuación de Schrödinger de teoría de campo también requiere regularización, para hacer que los operadores integrales sean finitos. El truncamiento tradicional del sistema en el espacio de Fock, donde el número permitido de partículas es limitado, normalmente interrumpe la regularización al eliminar infinitas partes que de otro modo se cancelarían frente a las partes retenidas. Aunque hay formas de evitar esto, no son completamente satisfactorias.
El método LFCC evita estas dificultades truncando el conjunto de ecuaciones de una manera muy diferente. En lugar de truncar el número de partículas, trunca la forma en que las funciones de onda se relacionan entre sí; Las funciones de onda de los estados de Fock superiores están determinadas por las funciones de onda de los estados inferiores y la exponenciación de un operador.. Específicamente, el eigenstate se escribe en la forma, dónde es un factor de normalización y es un estado con el número mínimo de componentes. El operadoraumenta el número de partículas y conserva todos los números cuánticos relevantes, incluido el momento del frente de luz. Esto es en principio exacto pero también infinito, porquepuede tener un número infinito de términos. Los modos cero pueden incluirse mediante la inclusión de su creación como términos en; esto genera un vacío no trivial como un estado coherente generalizado de modos cero.
El truncamiento realizado es un truncamiento de . El problema de valores propios original se convierte en un problema de valores propios de tamaño finito para el estado de valencia , combinado con ecuaciones auxiliares para los términos retenidos en :
Aquí es una proyección sobre el sector de valencia, y es el hamiltoniano efectivo de la LFCC. La proyección se trunca para proporcionar suficientes ecuaciones auxiliares para determinar las funciones en el truncado operador. El hamiltoniano efectivo se calcula a partir de su expansión de Baker-Hausdorff , which can be terminated at the point where more particles are being created than are kept by the truncated projection . The use of the exponential of rather than some other function is convenient, not only because of the Baker—Hausdorff expansion but more generally because it is invertible; in principle, other functions could be used and would also provide an exact representation until a truncation is made.
The truncation of can be handled systematically. Terms can be classified by the number of annihilated constituents and the net increase in particle number. For example, in QCD the lowest-order contributions annihilate one particle and increase the total by one. These are one-gluon emission from a quark, quark pair creation from one gluon, and gluon pair creation from one gluon. Each involves a function of relative momentum for the transition from one to two particles. Higher order terms annihilate more particles and/or increase the total by more than one. These provide additional contributions to higher-order wave functions and even to low-order wave functions for more complicated valence states. For example, the wave function for the Fock state of a meson can have a contribution from a term in that annihilates a pair and creates a pair plus a gluon, when this acts on the meson valence state .
The mathematics of the LFCC method has its origin in the many-body coupled cluster method used in nuclear physics and quantum chemistry.[27] The physics is, however, quite different. The many-body method works with a state of a large number of particles and uses the exponentiation of to build in correlations of excitations to higher single-particle states; the particle number does not change. The LFCC method starts from a small number of constituents in a valence state and uses to build states with more particles; the method of solution of the valence-state eigenvalue problem is left unspecified.
The computation of physical observables from matrix elements of operators requires some care. Direct computation would require an infinite sum over Fock space. One can instead borrow from the many-body coupled cluster method[27] a construction that computes expectation values from right and left eigenstates. This construction can be extended to include off-diagonal matrix elements and gauge projections. Physical quantities can then be computed from the right and left LFCC eigenstates.
Grupo de renormalización
Renormalization concepts, especially the renormalization group methods in quantum theories and statistical mechanics, have a long history and a very broad scope. The concepts of renormalization that appear useful in theories quantized in the front form of dynamics are essentially of two types, as in other areas of theoretical physics. The two types of concepts are associated with two types of theoretical tasks involved in applications of a theory. One task is to calculate observables (values of operationally defined quantities) in a theory that is unambiguously defined. The other task is to define a theory unambiguously. This is explained below.
Since the front form of dynamics aims at explaining hadrons as bound states of quarks and gluons, and the binding mechanism is not describable using perturbation theory, the definition of a theory needed in this case cannot be limited to perturbative expansions. For example, it is not sufficient to construct a theory using regularization of loop integrals order-by-order and correspondingly redefining the masses, coupling constants, and field normalization constants also order-by-order. In other words, one needs to design the Minkowski space-time formulation of a relativistic theory that is not based on any a priori perturbative scheme. The front form of Hamiltonian dynamics is perceived by many researchers as the most suitable framework for this purpose among the known options.[1][2][3]
The desired definition of a relativistic theory involves calculations of as many observables as one must use in order to fix all the parameters that appear in the theory. The relationship between the parameters and observables may depend on the number of degrees of freedom that are included in the theory.
For example, consider virtual particles in a candidate formulation of the theory. Formally, special relativity requires that the range of momenta of the particles is infinite because one can change the momentum of a particle by an arbitrary amount through a change of frame of reference. If the formulation is not to distinguish any inertial frame of reference, the particles must be allowed to carry any value of momentum. Since the quantum field modes corresponding to particles with different momenta form different degrees of freedom, the requirement of including infinitely many values of momentum means that one requires the theory to involve infinitely many degrees of freedom. But for mathematical reasons, being forced to use computers for sufficiently precise calculations, one has to work with a finite number of degrees of freedom. One must limit the momentum range by some cutoff.
Setting up a theory with a finite cutoff for mathematical reasons, one hopes that the cutoff can be made sufficiently large to avoid its appearance in observables of physical interest, but in local quantum field theories that are of interest in hadronic physics the situation is not that simple. Namely, particles of different momenta are coupled through the dynamics in a nontrivial way, and the calculations aiming at predicting observables yield results that depend on the cutoffs. Moreover, they do so in a diverging fashion.
There may be more cutoff parameters than just for momentum. For example, one may assume that the volume of space is limited, which would interfere with translation invariance of a theory, or assume that the number of virtual particles is limited, which would interfere with the assumption that every virtual particle may split into more virtual particles. All such restrictions lead to a set of cutoffs that becomes a part of a definition of a theory.
Consequently, every result of a calculation for any observable characterized by its physical scale has the form of a function of the set of parameters of the theory, , the set of cutoffs, say , and the scale . Thus, the results take the form
However, experiments provide values of observables that characterize natural processes irrespective of the cutoffs in a theory used to explain them. If the cutoffs do not describe properties of nature and are introduced merely for making a theory computable, one needs to understand how the dependence on may drop out from . The cutoffs may also reflect some natural features of a physical system at hand, such as in the model case of an ultraviolet cutoff on the wave vectors of sound waves in a crystal due to the spacing of atoms in the crystal lattice. The natural cutoffs may be of enormous size in comparison to the scale . Then, one faces the question of how it happens in the theory that its results for observables at scale are not also of the enormous size of the cutoff and, if they are not, then how they depend on the scale .
The two types of concepts of renormalization mentioned above are associated with the following two questions:
- How should the parameters depend on the cutoffs so that all observables of physical interest do not depend on , including the case where one removes the cutoffs by sending them formally to infinity?
- What is the required set of parameters ?
The renormalization group concept associated with the first question[28][29] predates the concept associated with the second question.[30][31][32][33] Certainly, if one were in possession of a good answer to the second question, the first question could also be answered. In the absence of a good answer to the second question, one may wonder why any specific choice of parameters and their cutoff dependence could secure cutoff independence of all observables with finite scales .
The renormalization group concept associated with the first question above relies on the circumstance that some finite set yields the desired result,
In this way of thinking, one can expect that in a theory with parameters a calculation of observables at some scale is sufficient to fix all parameters as functions of . So, one may hope that there exists a collection of effective parameters at scale , corresponding to observables at scale , that are sufficient to parametrize the theory in such a way that predictions expressed in terms of these parameters are free from dependence on . Since the scale is arbitrary, a whole family of such -parameter sets labeled by should exist, and every member of that family corresponds to the same physics. Moving from one such family to another by changing one value of to another is described as action of the renormalization group. The word group is justified because the group axioms are satisfied: two such changes form another such change, one can invert a change, etc.
The question remains, however, why fixing the cutoff dependence of parameters on , using conditions that selected observables do not depend on , is good enough to make all observables in the physical range of not depend on . In some theories such a miracle may happen but in others it may not. The ones where it happens are called renormalizable, because one can normalize the parameters properly to obtain cutoff independent results.
Typically, the set is established using perturbative calculations that are combined with models for description of nonperturbative effects. For example, perturbative QCD diagrams for quarks and gluons are combined with the parton models for description of binding of quarks and gluons into hadrons. The set of parameters includes cutoff dependent masses, charges and field normalization constants. The predictive power of a theory set up this way relies on the circumstance that the required set of parameters is relatively small. The regularization is designed order-by-order so that as many formal symmetries as possible of a local theory are preserved and employed in calculations, as in the dimensional regularization of Feynman diagrams. The claim that the set of parameters leads to finite, cutoff independent limits for all observables is qualified by the need to use some form of perturbation theory and inclusion of model assumptions concerning bound states.
The renormalization group concept associated with the second question above is conceived to explain how it may be so that the concept of renormalization group associated with the first question can make sense, instead of being at best a successful recipe to deal with divergences in perturbative calculations.[34] Namely, to answer the second question, one designs a calculation (see below) that identifies the required set of parameters to define the theory, the starting point being some specific initial assumption, such as some local Lagrangian density which is a function of field variables and needs to be modified by including all the required parameters. Once the required set of parameters is known, one can establish a set of observables that are sufficient to define the cutoff dependence of the required set. The observables can have any finite scale , and one can use any scale to define the parameters , up to their finite parts that must be fitted to experiment, including features such as the observed symmetries.
Thus, not only the possibility that a renormalization group of the first type may exist can be understood, but also the alternative situations are found where the set of required cutoff dependent parameters does not have to be finite. Predictive power of latter theories results from known relationships among the required parameters and options to establish all the relevant ones.[35]
The renormalization group concept of the second kind is associated with the nature of the mathematical computation used to discover the set of parameters . In its essence, the calculation starts with some specific form of a theory with cutoff and derives a corresponding theory with a smaller cutoff, in the sense of more restrictive, say . After re-parameterization using the cutoff as a unit, one obtains a new theory of similar type but with new terms. This means that the starting theory with cutoff should also contain such new terms for its form to be consistent with the presence of a cutoff. Eventually, one can find a set of terms that reproduces itself up to changes in the coefficients of the required terms. These coefficients evolve with the number of steps one makes, in each and every step reducing the cutoff by factor of two and rescaling variables. One could use other factors than two, but two is convenient.
In summary, one obtains a trajectory of a point in a space of dimension equal to the number of required parameters and motion along the trajectory is described by transformations that form new kind of a group. Different initial points might lead to different trajectories, but if the steps are self-similar and reduce to a multiple action of one and the same transformation, say , one may describe what happens in terms of the features of , called the renormalization group transformation. The transformation may transform points in the parameter space making some of the parameters decrease, some grow, and some stay unchanged. It may have fixed points, limit cycles, or even lead to chaotic motion.
Suppose that has a fixed point. If one starts the procedure at this point, an infinitely long sequence of reductions of the cutoff by factors of two changes nothing in the structure of the theory, except the scale of its cutoff. This means that the initial cutoff can be arbitrarily large. Such a theory may possess the symmetries of special relativity, since there is no price to pay for extending the cutoff as required when one wishes to make the Lorentz transformation that yields momenta which exceed the cutoff.
Both concepts of the renormalization group can be considered in quantum theories constructed using the front form of dynamics. The first concept allows one to play with a small set of parameters and seek consistency, which is a useful strategy in perturbation theory if one knows from other approaches what to expect. In particular, one may study new perturbative features that appear in the front form of dynamics, since it differs from the instant form. The main difference is that the front variables (or ) are considerably different from the transverse variables (or ), so that there is no simple rotational symmetry among them. One can also study sufficiently simplified models for which computers can be used to carry out calculations and see if a procedure suggested by perturbation theory may work beyond it. The second concept allows one to address the issue of defining a relativistic theory ab initio without limiting the definition to perturbative expansions. This option is particularly relevant to the issue of describing bound states in QCD. However, to address this issue one needs to overcome certain difficulties that the renormalization group procedures based on the idea of reduction of cutoffs are not capable of easily resolving. To avoid the difficulties, one can employ the similarity renormalization group procedure. Both the difficulties and similarity are explained in the next section.
Transformaciones de similitud
A glimpse of the difficulties of the procedure of reducing a cutoff to cutoff in the front form of Hamiltonian dynamics of strong interactions can be gained by considering the eigenvalue problem for the Hamiltonian ,
where , has a known spectrum and describes the interactions. Let us assume that the eigenstate can be written as a superposition of eigenstates of and let us introduce two projection operators, and , such that projects on eigenstates of with eigenvalues smaller than and projects on eigenstates of with eigenvalues between and . The result of projecting the eigenvalue problem for using and is a set of two coupled equations
The first equation can be used to evaluate in terms of ,
This expression allows one to write an equation for in the form
where
The equation for appears to resemble an eigenvalue problem for . It is valid in a theory with cutoff , but its effective Hamiltonian depends on the unknown eigenvalue . However, if is much greater than of interest, one can neglect in comparison to provided that is small in comparison to .
In QCD, which is asymptotically free, one indeed has as the dominant term in the energy denominator in for small eigenvalues . In practice, this happens for cutoffs so much larger than the smallest eigenvalues of physical interest that the corresponding eigenvalue problems are too complex for solving them with required precision. Namely, there are still too many degrees of freedom. One needs to reduce cutoffs considerably further. This issue appears in all approaches to the bound state problem in QCD, not only in the front form of the dynamics. Even if interactions are sufficiently small, one faces an additional difficulty with eliminating -states. Namely, for small interactions one can eliminate the eigenvalue from a proper effective Hamiltonian in -subspace in favor of eigenvalues of . Consequently, the denominators analogous to the one that appears above in only contain differences of eigenvalues of , one above and one below.[30][31] Unfortunately, such differences can become arbitrarily small near the cutoff , and they generate strong interactions in the effective theory due to the coupling between the states just below and just above the cutoff . This is particularly bothersome when the eigenstates of near the cutoff are highly degenerate and splitting of the bound state problem into parts below and above the cutoff cannot be accomplished through any simple expansion in powers of the coupling constant.
In any case, when one reduces the cutoff to , and then to and so on, the strength of interaction in QCD Hamiltonians increases and, especially if the interaction is attractive, can cancel and cannot be ignored no matter how small it is in comparison to the reduced cutoff. In particular, this difficulty concerns bound states, where interactions must prevent free relative motion of constituents from dominating the scene and a spatially compact systems have to be formed. So far, it appears not possible to precisely eliminate the eigenvalue from the effective dynamics obtained by projecting on sufficiently low energy eigenstates of to facilitate reliable calculations.
Fortunately, one can use instead a change of basis.[36] Namely, it is possible to define a procedure in which the basis states are rotated in such a way that the matrix elements of vanish between basis states that according to differ in energy by more than a running cutoff, say . The running cutoff is called the energy bandwidth. The name comes from the band-diagonal form of the Hamiltonian matrix in the new basis ordered in energy using . Different values of the running cutoff correspond to using differently rotated basis states. The rotation is designed not to depend at all on the eigenvalues one wants to compute.
As a result, one obtains in the rotated basis an effective Hamiltonian matrix eigenvalue problem in which the dependence on cutoff may manifest itself only in the explicit dependence of matrix elements of the new .[36] The two features of similarity that (1) the -dependence becomes explicit before one tackles the problem of solving the eigenvalue problem for and (2) the effective Hamiltonian with small energy bandwidth may not depend on the eigenvalues one tries to find, allow one to discover in advance the required counterterms to the diverging cutoff dependence. A complete set of counterterms defines the set of parameters required for defining the theory which has a finite energy bandwidth and no cutoff dependence in the band. In the course of discovering the counterterms and corresponding parameters, one keeps changing the initial Hamiltonian. Eventually, the complete Hamiltonian may have cutoff independent eigenvalues, including bound states.
In the case of the front-form Hamiltonian for QCD, a perturbative version of the similarity renormalization group procedure is outlined by Wilson et al.[37] Further discussion of computational methods stemming from the similarity renormalization group concept is provided in the next section.
Procedimiento de grupo de renormalización para partículas efectivas
The similarity renormalization group procedure, discussed in #Similarity transformations, can be applied to the problem of describing bound states of quarks and gluons using QCD according to the general computational scheme outlined by Wilson et al.[37] and illustrated in a numerically soluble model by Glazek and Wilson.[38] Since these works were completed, the method has been applied to various physical systems using a weak-coupling expansion. More recently, similarity has evolved into a computational tool called the renormalization group procedure for effective particles, or RGPEP. In principle, the RGPEP is now defined without a need to refer to some perturbative expansion. The most recent explanation of the RGPEP is given by Glazek in terms of an elementary and exactly solvable model for relativistic fermions that interact through a mass mixing term of arbitrary strength in their Hamiltonian.[39][40]
The effective particles can be seen as resulting from a dynamical transformation akin to the Melosh transformation from current to constituent quarks.[41] Namely, the RGPEP transformation changes the bare quanta in a canonical theory to the effective quanta in an equivalent effective theory with a Hamiltonian that has the energy bandwidth ; see #Similarity transformations and references therein for an explanation of the band. The transformations that change form a group.
The effective particles are introduced through a transformation
where is a quantum field operator built from creation and annihilation operators for effective particles of size and is the original quantum field operator built from creation and annihilation operators for point-like bare quanta of a canonical theory. In great brevity, a canonical Hamiltonian density is built from fields and the effective Hamiltonian at scale is built from fields , but without actually changing the Hamiltonian. Thus,
which means that the same dynamics is expressed in terms of different operators for different values of . The coefficients in the expansion of a Hamiltonian in powers of the field operators depend on and the field operators depend on , but the Hamiltonian is not changing with . The RGPEP provides an equation for the coefficients as functions of .
In principle, if one had solved the RGPEP equation for the front form Hamiltonian of QCD exactly, the eigenvalue problem could be written using effective quarks and gluons corresponding to any . In particular, for very small, the eigenvalue problem would involve very large numbers of virtual constituents capable of interacting with large momentum transfers up to about the bandwidth . In contrast, the same eigenvalue problem written in terms of quanta corresponding to a large , comparable with the size of hadrons, is hoped to take the form of a simple equation that resembles the constituent quark models. To demonstrate mathematically that this is precisely what happens in the RGPEP in QCD is a serious challenge.
Ecuación de Bethe-Salpeter
The Bethe-Salpeter amplitude, which satisfies the Bethe-Salpeter equation[42][43][44] (see the reviews by Nakanishi[45][46] ), when projected on the light-front plane, results in the light-front wave function. The meaning of the ``light-front projection" is the following. In the coordinate space, the Bethe-Salpeter amplitude is a function of two four-dimensional coordinates , namely: , where is the total four-momentum of the system. In momentum space, it is given by the Fourier transform:
(the momentum space Bethe-Salpeter amplitude defined in this way includes in itself the delta-function responsible for the momenta conservation ). The light-front projection means that the arguments are on the light-front plane, i.e., they are constrained by the condition (in the covariant formulation): . This is achieved by inserting in the Fourier transform the corresponding delta functions :
In this way, we can find the light-front wave function . Applying this formula to the Bethe-Salpeter amplitude with a given total angular momentum, one reproduces the angular momentum structure of the light-front wave function described in Light front quantization#Angular momentum. In particular, projecting the Bethe-Salpeter amplitude corresponding to a system of two spinless particles with the angular momentum , one reproduces the light-front wave function
given in Light front quantization#Angular momentum.
The Bethe-Salpeter amplitude includes the propagators of the external particles, and, therefore, it is singular. It can be represented in the form of the Nakanishi integral[47] through a non-singular function :
(1)
where is the relative four-momentum. The Nakanishi weight function is found from an equation and has the properties: , . Projecting the Bethe-Salpeter amplitude (1) on the light-front plane, we get the following useful representation for the light-front wave function (see the review by Carbonell and Karmanov[48]):
It turns out that the masses of a two-body system, found from the Bethe-Salpeter equation for and from the light-front equation for with the kernel corresponding to the same physical content, say, one-boson exchange (which, however, in the both approaches have very different analytical forms) are very close to each other. The same is true for the electromagnetic form factors[49] This undoubtedly proves the existence of three-body forces, though the contribution of relativistic origin does not exhaust, of course, all the contributions. The same relativistic dynamics should generate four-body forces, etc. Since in nuclei the small binding energies (relative to the nucleon mass) result from cancellations between the kinetic and potentials energies (which are comparable with nucleon mass, and, hence relativistic), the relativistic effects in nuclei are noticeable. Therefore, many-body forces should be taken into account for fine tuning to experimental data.
Estructura de vacío y modos cero.
One of the advantages of light-front quantization is that the empty state, the so-called perturbative vacuum, is the physical vacuum.[50][51][52][53][54][55][56][57][58][59][60] The massive states of a theory can then be built on this lowest state without having any contributions from vacuum structure, and the wave functions for these massive states do not contain vacuum contributions. This occurs because each is positive, and the interactions of the theory cannot produce particles from the zero-momentum vacuum without violating momentum conservation. There is no need to normal-order the light-front vacuum.
However, certain aspects of some theories are associated with vacuum structure. For example, the Higgs mechanism of the Standard Model relies on spontaneous symmetry breaking in the vacuum of the theory.[61][62][63][64][65][66] The usual Higgs vacuum expectation value in the instant form is replaced by zero mode analogous to a constant Stark field when one quantizes the Standard model using the front form.[67] Chiral symmetry breaking of quantum chromodynamics is often associated in the instant form with quark and gluon condensates in the QCD vacuum. However, these effects become properties of the hadron wave functions themselves using the front form.[59][60][68][69] This also eliminates the many orders of magnitude conflict between the measured cosmological constant and quantum field theory.[68]
Some aspects of vacuum structure in light-front quantization can be analyzed by studying properties of massive states. In particular, by studying the appearance of degeneracies among the lowest massive states, one can determine the critical coupling strength associated with spontaneous symmetry breaking. One can also use a limiting process, where the analysis begins in equal-time quantization but arrives in light-front coordinates as the limit of some chosen parameter.[70][71] A much more direct approach is to include modes of zero longitudinal momentum (zero modes) in a calculation of a nontrivial light-front vacuum built from these modes; the Hamiltonian then contains effective interactions that determine the vacuum structure and provide for zero-mode exchange interactions between constituents of massive states.
Ver también
- Light front quantization
- Light-front quantization applications
- Quantum field theories
- Quantum chromodynamics
- Quantum electrodynamics
- Light-front holography
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enlaces externos
- ILCAC, Inc., the International Light-Cone Advisory Committee.
- Publications on light-front dynamics, maintained by A. Harindranath.