Abstract
We investigate two examples of conformal invariant pure spinor fermionic models, which admit particle-like solutions of the classical field equations. For different dimensions and quantum spinor numbers, the vector field visualizations of the models are constructed to provide a better understanding of the spinor-type instanton dynamics in phase space. The hierarchical cluster analysis method investigations of the models are also presented. Finally, the autocorrelation and power spectrum graphs of models are constructed and frequencies of motions are defined.
1. Introduction
The success of electron and antielectron (positron) interpretation of Dirac’s nonlinear spinor field wave equation brought with itself a great effort among theoretical physicists in searching new nonlinear field equations. Heisenberg spent his years to formulate a “theory of everything” using only fermions [1]. A few years later another attempt in this direction came with the work of Gursey [2]. Gursey proposed his model as a possible basis for a unitary description of elementary particles to remedy some of the problems the Heisenberg model had. Gursey model is the first conformally invariant pure fermionic model and it is four-dimensional [2]. It possesses broader dynamical symmetries compared to Dirac’s and Heisenberg et al.’s works. What is more important is that Gursey’s model is suitable for extensions to other particles having spin. In the ensuing years, Kortel found a class of exact solutions of Gursey spinor wave equation and these were shown to be instantonic and meronic in character [3, 4]. After the spinor-type instanton solutions of this model with nonlinear self-coupled spinor term have been found by the spontaneous symmetry breaking of the conformal invariance of , that is, [4], many works have been done on this model [5–9]. Gursey model is very important because of the similarity of these solutions to the solutions of pure Yang-Mills theories in four dimensions.
Another fermionic model is the Thirring model which describes Dirac fermions in (1 + 1) space-time dimensions with the nonlinear self-interaction term [10]. Thirring model is also conformally invariant and with no mass term. It played a very fruitful role in the progress of quantum field theory [10]. The spinor-type instanton solutions of Thirring model were also found via breaking of conformal symmetry and were shown that these solutions were stable [11]. This model is simple enough to be analyzed with full details; and yet it contains many of the typical features of the quantization of relativistic quantum field theories [12].
It is well known that instantons are classical topological solutions in quantum field theories which lie behind the standard model of particles. The aim of this study is to provide a better understanding to the dynamics of spinor-type instanton solutions of fermionic models in phase space for different dimensions and quantum spinor numbers. For this purpose, the vector fields of conformal invariant pure spinor fermionic models are constructed and their Jacobian determinants are calculated. In addition, the autocorrelations and power spectra [13] are presented to see whether the oscillations are periodic or not. In the periodic oscillation case, the oscillation frequencies of motions are calculated. Moreover, hierarchical cluster analysis [14] method investigations of the fermionic models are presented.
2. Models
Below we present firstly the behavior of spinor-type instantons in Thirring model because of the lower space-time dimensionality and later do the same for Gursey model.
2.1. Thirring Model
In this model the two-dimensional fermion-fermion interactions are described by the conformal invariant Lagrangian where is the positive coupling constant and the fermion field has scale dimension [10]. The equation of motion that follows is
The complex form of the Euclidian configuration of Heisenberg ansatz [1] is with an arbitrary spinor constant, . , and are real functions of (, ). Inserting (3) into (2) we get where , since
By writing and , with and , , and [3], we achieved the dimensionless form of the nonlinear ordinary coupled differential equation system (4a) and (4b) as
Here and are dimensionless functions of and , are constants [3]. We call the equation system in (6a) and (6b) as the Thirring nonlinear differential equation system (TNDES). The solution of TNDES for is the Thirring instantons given in [11]. Recently, the role of the coupling constant in the evolution of Thirring instantons in phase space has been investigated [8, 15].
2.2. Gursey Model
In this 4-dimensional model the Lagrangian is based on fermion fields of scale dimension [2]
The equation of motion that follows is
The same ansatz (see (3)) is also used to solve (8) [3]. Here (, , ; ). Inserting (3) into (8) we getwhere with
Substituting and , with , , , and [3], we achieved the dimensionless form of the nonlinear ordinary coupled differential equation systems (9a) and (9b) as
We call this equation system the Gursey nonlinear differential equation system (GNDES). The solution of GNDES for is the Gursey instantons given in [4]. Very recently, the role of the coupling constant in the evolution of Gursey instantons in phase space has been investigated [8, 9]. For a better understanding of the quantum dynamics of spinor-type instantons in vacuum, the dynamics of Forced Gursey nonlinear differential equation system have been studied [8, 16].
3. Vector Analysis of the Models
TNDES can be written in the form of a vector field [17] as follows by defining a new constant since
Due to the Liouville theorem TNDES is conserved, as is also seen from the vector field displays given in Figure 1.
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On the other hand, the Jacobian of TNDES is
In Figure 1(a), the vector field of TNDES is displayed for . Figures 1(b)–1(d) show the vector field display of TNDES for , , and 1.5, respectively. is the special case; instanton type solutions, called “Thirring instantons,” have been found [11]. Figure 2 shows behavior versus and for . is clearly positive outside of a tiny neighbor of in which it is negative; therefore, the flow is orientation preserving (Figure 1(c)). The system is conservative along the flow.
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The characteristic eigenvalues providing an understanding of the topological behavior around the singularity points come from as
Table 1 presents the fix points, their eigenvalues, and characterizations at three values in the interval. We replaced fix points into (15) to find the behavior of the system around the fix points. For , one finds reel eigenvalues. Hence the fix point is hyperbolic, as is seen in Figure 1(a). For the other values, eigenvalues are purely imaginary. So the fix points are elliptic. These results are compatible with the phase space displays seen in Figure 1.
Analogously, GNDES can be written in the form of a vector field as follows, by defining a new constant :
We take the divergence of
Density is conserved since ; so phase space volumes are conserved as in Figure 3. We find the Jacobian of GNDES as
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Alike, the vector fields of GNDES are shown in Figures 3(a)–3(d) and the evolution of versus and is shown in Figure 4 for . For , instanton type solutions called “Gursey instantons” have been found [4]. In comparison with Figures 1(a) and 3(a), the vector visualization of Gursey instantons has the same behavior in phase space like Thirring instantons. The Gursey flow is also conservative.
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In this case the characteristic eigenvalues come from as
Table 2 presents the fix points, their eigenvalues, and characterizations for five values in the interval. For , one finds reel eigenvalues. Hence the fix point is hyperbolic, as seen in Figure 3(a). For the other values, eigenvalues are purely imaginary. So the fix points are elliptic as in the Thirring case. These results are compatible with the phase space displays seen in Figure 3.
4. Cluster Analysis of the Models
As additional information, we study the models using the hierarchical cluster analysis method. Hierarchical methods usually produce a graphical output known as a dendrogram graph that shows the hierarchical clustering structure [14]. In Figure 5, the dendrogram graphs belonging to hierarchical clustering of time series for TNDES solutions for , , and are shown. In these graphs, the -axis represents the similarity or correlation percentages (%0–%100) belonging to our data and increases towards right. Along the -axis the fusion of clusters due to similarities of data is recorded. For each value we plotted the dendrogram graphs for all data and give only the truncate dendrogram graphs which are the summary of the first 20 mergers having the same similarity percentage. From hierarchical point of view, for all values clustering is gathered in two different main groups. However, for there are five different clusterings in the first main cluster; in the second main cluster there are ten different clusterings and for there are six in the first main cluster. In the second main cluster there are twelve different clusterings. Finally, for , there are seven different clusterings in the first main cluster and only twelve different clusterings in the second main one.
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On the other hand, the dendrogram graphs belonging to the hierarchical clustering of time series for GNDES solutions for , , and are shown in Figure 6. Analogously for each value we plotted the dendrogram graphs for all data values and give only the truncate dendrogram graphs which are the summary of the first 20 mergers having the same similarity percentage. For all values, clustering is gathered in two different main groups comparably TNDES. However, for there are four different clusterings in the first main cluster; in the second main cluster there are eleven different clusterings and for in the first main cluster there are seven different clusterings and in the second main cluster there are fourteen different clusterings as distinct from TNDES. Besides for , there are seven different clusterings in the first main cluster and only thirteen different clusterings in the second main one.
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5. Autocorrelations and Power Spectrums of the Models
Autocorrelation gives similarities at two points in a time series as a function of how separated the points in time [13] is and it is defined by
is the correlation length or time interval. We show the autocorrelation plots depending on different and parameters in Figure 7. The autocorrelation plots have the same periods with time series of the solutions of GNDES and TNDES . For each and parameters, the motions are periodic in time. When the nonlinearity increases, period of the oscillation gets smaller.
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The relation between the oscillation frequency and autocorrelation is defined by power spectrum. The power spectrum is the Fourier transform of the autocorrelation function:
The power spectrum shows the energy distribution [13] over different frequencies of systems . Power spectrum plots describe the periodic oscillation as the same as autocorrelation plots. However the solution is almost sinusoidal but not exactly. Therefore, in power spectrums there is a fundamental frequency and there are some small harmonics close to the fundamental one. In Figure 8(a) we show the power spectrum of TNDES for various values. For , and its harmonics are 1.15 and 1.916, for , and its harmonic is 2.376, and finally for , . In Figure 8(b) we show the power spectrum of GNDES for various values. For , and its harmonics are 0.6952, 1.05, 1.399, and 1.751, for , and its harmonic is 1.399, and finally for , and its harmonic is 2.262.
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6. Conclusion
In this paper we focus on the vector analysis and orientations of GNDES and TNDES to provide a better understanding of the behaviors of spinor-type instanton solutions of fermionic models in phase space for different dimensions and quantum spinor numbers. For this purpose the vector fields of four-dimensional Gursey instantons () [4] and two-dimensional Thirring instantons () [11] are constructed and it is shown that the systems are conservative along the flow. The situation is similar for the other and values. In order to get more information on the orientation of systems in phase space, Jacobeans are calculated and plotted versus and . This gives us an idea about whether the orientation of phase space area is preserved or not. The orientations of the four-dimensional Gursey instantons are similar to two-dimensional Thirring instantons in phase space.
Characterization of the fix points of systems are also briefly presented by calculating the characteristic eigenvalues. All eigenvalues are purely imaginary for both models so the fix points are elliptic except for and . For this parameter value, the fix points are hyperbolic as can be seen in Figures 1 and 3.
As additional information, we study systems using the hierarchical cluster analysis method. Similarities in time series are displayed in Figures 5 and 6 by using the hierarchical cluster analysis method. As the result, we can say that the similarity of data increases depending on the increment of the nonlinear control parameter for both models.
Finally we investigate autocorrelation and power spectrum of the systems with respect to the control parameters. It will be remarkable to notice that, as seen from Figure 7, the motions of systems are periodic for each value of the control parameters and as control parameters increase, periods of the oscillations get smaller and so frequency gets bigger for both models. Benefits from Fourier transform of the autocorrelation, fundamental frequency, and harmonics are calculated. As a result of our calculations we can say that the solution is almost sinusoidal but not exactly the same. The systems have the same behaviors as the undamped harmonic oscillator.
In the view of obtained results, we can interestingly conclude that although models have different quantum spinor numbers and dimensions, the behaviors of four-dimensional spinor-type Gursey instantons are similar to those of two-dimensional spinor-type Thirring instantons. This result is compatible with the knowledge of the critical behavior of a conformal symmetric quantum field theory near the fixed points [18–20]. Conformal symmetry fixes the 2-point and 3-point correlation functions of primary fields since the only difference between two models is constants that depend on the interaction constant and dimensions and other details of the models. Hence the classical solutions should have similar behaviors [18–24]. So we confirm that the behaviors of spinor-type instantons in phase space are not dependent on the quantum fractional spinor number as well as dimensions [8, 9].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank Professor Dr. K. Gediz Akdeniz and Professor Dr. Emine Rizaoglu for their support while preparing this paper. This work was supported by Scientific Research Projects Coordination Unit of Istanbul University, Project nos. 33111 and 35737.