A general procedure to find ground state solutions for finite M(atrix) theory. Reduced models and SUSY quantum cosmology
We propose a general method to find exact ground state solutions to the SU() invariant matrix model arising from the quantization of the 11-dimensional supermembrane action in the light-cone gauge. We illustrate the method by applying it to lower dimensional models and for the SU(2) group. This approach can be used to find ground state solutions to the complete 9-dimensional model and for any SU() group. The supercharges and the constraints related to the SU() symmetry are the relevant operators and they generate a multicomponent wave function. In the procedure, the fermionic degrees of freedom are represented by means of Dirac-like gamma matrices. We exhibit a relation between these finite matrix theory ground state solutions and SUSY quantum cosmology wave functions giving a possible physical significance to the theory even for finite
pacs:11.25.-w, 04.60.Ds, 04.65.+e, 04.60.Kz, 12.60.Jv
As is known, M-theory should be an eleven dimensional supersymmetric quantum theory from which all superstring theories can be deduced and it can be connected also to 11-dimensional supergravity. This idea was born since it was realized that the five consistent theories of strings and 11- supergravity are related among them through several types of dualities 1.1 (); 1.2 (); 1.3 (); 1.4 (); 1.5 (). There are interesting approaches and calculations that provide the possibility to understand some important features of this fundamental theory. There is a concrete formulation of M-theory which has been claimed has the appropriate expected properties and it is based on the matrix model conjectured in (BFSS 1.1 ()). This matrix model is a supersymmetric quantum mechanics where the bosonic degrees of freedom are a finite set of Hermitian matrices Sato (). We can associate a time dependent matrix to every point in space and these matricial coordinates are obviously noncommuting. The existence of dynamical objects of dimensionalities higher than one, like -branes, has left aside the fundamental status of particles and strings and the origin of this matrix model and its connection to M-theory arises from the dynamics of D0 branes Polchinski (); Witten (); Danielsson (). It was conjectured in 1.1 () that M-theory is the limit of the maximally supersymmetric matrix Hamiltonian emerging from the D0-brane matrix model. The conjecture was extended in 1.6 () showing that matrix theory is also meaningful for finite , so the solutions to the matrix model for finite are relevant.
In the recent years, research has been done on another matrix model arising from noncommutative gauge theories ikkt1 (), it is called the IKKT matrix model and it can also be related to a large reduced model of 10-dimensional non-commutative Super Yang-Mills theory. In the IKKT framework, the parameter that measures the non-commutativity between the coordinates of space-time becomes dynamical and in the semiclassical limit it defines a Poisson structure such that the metric of space-time itself emerges naturally stkr1 (); stkr2 (); stkr3 (). This matrix model can be seen as a noncommutative Super Yang-Mills theory on . The path integral quantization of this matrix model is naturally defined by integrating over the space of matrices and in a semiclassical way, it reproduces some interesting features of curved spacetime stkr4 (). So, it has been claimed is a good candidate to be a quantum theory of gravity together with other fundamental interactions stkr5 (). The presence of branes in this theory stkr6 () becomes also important as it is in the matrix theory of BFSS. The quantization approaches are different for the two matrix models briefly described above, but they have the same common D-brane world origin and both involve the most important characteristic features of a unified theory of physical interactions including gravity.
Another, and somehow independent, matrix model was encountered in the quantization of the classical 11-dimensional supermembrane action in the light cone gauge (LCG) N1 (); N2 (); N3 (); Halpern1 (). These matrix models were studied in the context of supersymmetric quantum mechanics years before the quantization of the supermembrane was finally achieved Halpern (). The action of the supermembrane is the generalization of the Green-Schwarz action for the superstring for a membrane moving in a -dimensional target superspace, and when this can be directly related to 11-dimensional supergravity Sezgin (). It is possible to regularize the theory because of the existence of a finite dimensional Hilbert space where the area preserving diffeomorphisms (APD) invariant states can be expressed. This APD invariance is translated into a SU() symmetry in the regularized theory which is equivalent to a Super Yang-Mills quantum mechanics matrix model N1 (); N2 (); N3 (). The Hamiltonian and the associated ground state of these matrix models of supersymmetric quantum mechanics with SU() invariance were studied in Halpern () and the matrix model arising from the supermembrane action is exactly one of those. In this type of supersymmetric models it is of great importance to find the ground state solution Halpern (); Halpern1 (); Sethi1 (); Sethi2 (), and for the matrix model we have been briefly mentioned here the existence of such state, even for finite, has been found only perturbatively.
These two matrix models, the one related to D0-branes and the other related to the supermembrane, are not independent. There is a connection between membranes and D0-branes first speculated in Town () where it was considered that membranes could be regarded as collective excitations of D0-branes. This connection was treated in great detail in 1.1 (). The noncommutative space defined by the phase space coordinates where the membranes exist has a basic minimum area, this quantum unit cells of space are precisely the D0-branes, each one with a longitudinal momentum of 1/. The longitudinal direction is the one compactified in the LCG description. This relation of duality connects the two matrix models.
The first purpose of this work is to exemplify the general method by finding solutions to the ground state of matrix theory reduced models arising from the quantization of the classical supermembrane action. Our procedure is based on; i) We consider finite matrix models, ii) The relevant operators are the supercharges and those related with the SU() symmetry. It is no more necessary to consider the Hamiltonian operator because through the canonical algebra it is a consequence of these constraints, iii) The fermionic degrees of freedom are represented utilizing a Dirac-like gamma matrix representation jefe1 (); jefe4 (). By means of these, the associated supercharges and the operators related to the SU() symmetry become matrix differential operators acting on a multicomponent wave function. One generates a Dirac-like quantum matrix model formalism.
With our proposal one can solve any finite matrix model and for all the bosonic and fermionic degrees of freedom. Even though finite matrix models are physically relevant 1.6 (), it is not clear which particular should be considered. For large , our method will generate a large number of first order coupled differential equations. In order to illustrate how to apply our proposal, we restrict ourselves to reduced models, for the SU() invariance group. Besides, for the sake of simplicity and as a way to demonstrate how this procedure works, we will make further assumptions in the space of bosonic variables. We will then freeze out some bosonic degrees of freedom. By doing this one cannot assure that this reduced models will capture all features of the quantization of finite matrix theory (“for a related discussion in the context of quantum cosmology see; Ryan ()”). Some of the models studied can however be exactly solved illustrating us about the kind of ground state solutions to be expected for a larger model. The method we propose can be directly extended to higher dimensional models and for any SU() group. This kind of representation for the fermionic variables has been used in a first and several models of SUSY quantum cosmology jefe1 (); jefe4 (); jefe3 (); jefe6 (); Torres (); SFmatter3 () leading to pretty much interesting features regarding every physical supersymmetric system treated in this way. The fermionic degrees of freedom can also be represented by differential operators jefe2 (); DEath1 (); Moniz (); PD2 (). A relevant point is that it is possible to solve the complete operators system containing both, the bosonic and the fermionic sectors. The fermionic information will then be encoded in each one of the entries of the multicomponent wave function and the physical information regarding the variables of the system will be given by means of the components of this wave function.
It has been argued that M-theory is a promising candidate to be a quantum theory of gravity, it is then expected that the matrix models related with it, contain quantum and classical gravitational information. The same authors of the conjecture have shown that a specific compactification of matrix theory correctly describes the properties of a Schwarzschild black hole 16.1 (); 16.2 (); 16.3 (); 16.4 () and other works related to matrix theory black holes and its thermodynamic properties are Bht1 (); Bht2 (); Bht3 (); Bht4 ().
It has also been claimed that classical cosmological models can be deduced directly from the matrix theory of BFSS Alvarez (); Gibbons () and specifically in Gibbons (), using a homothetic ansatz, the authors were able to derive the Friedmann equations from the bosonic sector of the classical Hamiltonian of matrix theory. Following this kind of ideas, the interesting point to us here is that we show that some supersymmetric cosmological models are then hidden in the SU() invariant supersymmetric matrix model as well. It is then expected to find supersymmetric quantum solutions that can be interpreted and related to SUSY quantum cosmology models. This is a particularly interesting byproduct of our reduced matrix models and shows that these reduced models are also supersymmetric given that their corresponding SUSY cosmological models are supersymmetric by construction jefe6 (); Torres (). Also recently Damour (), to describe the quantum dynamics of the supersymmetric Bianchi IX cosmological model jefe4 () it has been proposed to fix from the start the six degrees of freedom describing local Lorentz rotations of the tetrad. The operational content of the supercharges and Hamiltonian revealed a hidden hyperbolic Kac-Moody structure which seems to support the conjecture about a correspondence between supergravity and the dynamics of a spinning particle moving on an infinite dimensional coset space. At the same time we show that some reduced matrix models correspond to exact SUSY quantum cosmology models that arise from supergravity. All these relations point out to a web of connections between these different topics and make it of interest to search for a method to find ground state solutions in finite matrix theory models. Here, we present a general procedure to construct them. We will also be interested in relations between the solutions of these ground state matrix theory models and wave functions corresponding to SUSY quantum cosmology. One of the wave function solutions obtained in our reduced models resembles a wave function solution obtained in the superfield approach to supersymmetric quantum cosmology and it is then argued that the matrix model we consider provides the relevant physical information, as classical cosmology seems also to arise from bosonic matrix models Alvarez (); Gibbons ().
The work is organized as follows. First in section II we show the matrix model and its description as a supersymmetric SU() invariant quantum mechanical model. In section III we illustrate the way in which we look for solutions with the matricial representation for the fermionic degrees of freedom in a 2-dimensional reduced model and for the SU(2) group. In section IV we show the solutions we obtain for some particular assumptions on the bosonic variables. As we pointed out, this method could be directly extended to the complete 9-dimensional theory and for any SU() group, although for large powerful computational tools would be needed. Then in section V we make an extension to a 4 dimensional model. It is interesting to remark that some of the 2-dimensional solutions are also solutions to the 4-dimensional model. In section VI we show first that in this extension we encounter, exactly, a model guessed in N3 () and used to analyze the spectrum of the matrix Hamiltonian of the quantum supermembrane action and then we exhibit some solutions to this extended model. In section VII we discuss a connection of one of the solutions to the extended 4-dimensional model with SUSY quantum cosmology. Section VIII is devoted to conclusions.
Ii Matrix model
As we mentioned in the introduction, the model arising from the quantization of the classical supermembrane action, once the regularization is achieved, becomes a model of supersymmetric quantum mechanics 1.1 () that can also be obtained from a dimensional reduction of a super Yang-Mills theory with gauge group SU() Halpern (); N1 (), the Hamiltonian of this matrix model Halpern1 () is given by
where and are the bosonic degrees of freedom and its related momenta respectively, and are the components of the SU() valued matrices given by where are the basis group elements. Then is a group index and is a dimension index. represent the fermonic degrees of freedom and are Dirac matrices obeying the Clifford algebra . The SU() group structure constants are . The algebra between the supercharges as function of the total Hamiltonian and the supercharges themselves are given by
where . In the anticommutator of the supercharges it appears the operator (constraint) related to the SU() invariance which in terms of the bosonic and fermionic variables is given by
Now, we want to realize the quantization. And as we are working with a supersymmetric quantum mechanics where the Hamiltonian is the square of the supercharges and it is positive semi-definite, the state which obeys is automatically the ground state wave function. Also, the ground state should be a gauge invariant state, it should satisfy . The commutator and anticommutator relations for the bosonic and fermionic degrees of freedom will be
The supercharges arise as the square root of the Hamiltonian, and as it can be seen in the algebra eq. (2), we can solve the equations together with . Then the ground state satisfying this, will respect all the symmetries; gauge and exact supersymmetry. The representation of the canonical momenta corresponding to is, as usual, , and given the algebra for the fermionic variables eq. (4), we have the freedom to represent the as matrices. This kind of representation was used first in SUSY quantum cosmology jefe1 (). This is an essential tool jefe1 (); jefe3 (); jefe4 (); jefe6 () in this work and as we will show, we will be able to find exact solutions to some reduced lower dimensional models. The method we will present can be applied to search for solutions to the 9-dimensional supersymmetric matrix model and for any SU() group. However, as it will be seen, for these dimensions and a large this is a difficult task because one would need to solve an extremely large number of coupled partial differential equations.
Iii The method of solution
To begin with, we will try to find solutions to the ground state by solving the corresponding equations and (the Hamiltonian constraint is then satisfied) for a SU() symmetry; and for a 2-dimensional reduced model, . Hence, we will have six bosonic degrees of freedom and the same number for the fermionic ones. In the variables , is a group index and the index represents the number of supercharges which is in accordance with the number of components of a spinor in -dimensions. Then with , we have two supercharges and consequently six fermionic variables. The dimension of the square matrix representation depends on the number of fermionic degrees of freedom. So, in this first toy model . The representation we use for the matrices that satisfies eq. (4) is the following
where are the Pauli Matrices, are the Dirac matrices in a Majorana representation and is the identity matrix. The algebra in eq. (4) is not affected if we choose the plus, or minus, sign on every matrix in eq. (5). Notice that we have three kinds of indices, those corresponding to the supercharges , to the group , and the ones associated to the dimensions . Even when we have three kinds of indices we will denote all of them, as in eq. (5), with the same kind of number. As an example, the first of the matrices is explicitly
and for this model we choose the following two matrices
The two supercharges of eq. (2) are given by the following operators
and the three operators related with the SU(2) gauge symmetry are given by
Using the matrix representation of eq. (5), all of them are linear matrix differential operators acting on an eight component wave function . We can restrict this model even more, setting some of the bosonic variables equal to zero, or making some identifications between them. These will give specific configurations. Using the anticommutator of the supercharges in eq.(2) and eq. (7) we can see that it is not always necessary to impose all of the three operators, and that we can solve the problem for one single operator . For every , and , we have eight coupled partial differential equations for the eight components of the wave function . For example the equation leads to the eight equations
where the functions , depend on the components of the matrices and give rise to the potentials appearing in the Hamiltonian. These functions are given by
We can then make the mentioned simplifications to find a model with a particular potential, or even easier, for a null potential. This last is the first one for which we will find a solution.
Iv Exact Solutions
Here we will show the explicit form of the ground state wave function for a few models using some specific configurations of the components of the variables.
iv.1 Solution 1
The model is the following. We choose the bosonic variables to be (in units where )
and as a consequence of this choice we have
Hence, in this configuration there will be no bosonic potential and the operators take the simpler form
One should apply all the three operators to the wave function, however given eq. (2) and eq. (7) we have to consider only the operator acting on the wave function , and consistently it is sufficient to apply also one of the supercharges, namely . Relations between the eight components of arise and the wave function as a row vector has the form , where we mean by this that, , and so on. Because of the form of the differential equations for this two dimensional model, it is easier to solve them using polar coordinates . The exact solution for the two independent components of the wave function results in
Here is a separation constant and are arbitrary constants and we can choose them to have a normalizable solution.
iv.2 Solution 2
The following model is chosen in such a way that the bosonic part of the potential in its corresponding Hamiltonian takes the form of a one dimensional harmonic oscillator, this is
Following the same procedure as in the previous solution, we will have two independent components, and , for the wave function . The explicit form of these two components is
where are arbitrary constants, if we choose them to be , then we get a normalizable wave function.
iv.3 Solution 3
The next model has also a null potential, but it has three independent bosonic variables
We can see from eq. (2) and eq. (7) that for this particular configuration it is not necessary to impose any because , . We will try to find a solution to . This is possible and the eight independent components of the wave function reduce to only two, these are and , and the wave function would take the following form . Besides, there is a coupling between these two components, so they are also related. The form of the solution of is a linear combination of the exponential functions that come out from every particular choice of signs () in the exponential factor , where are constants and because of the coupling, results proportional to , so . The form of the constant depends on the particular form of . For instance and could be the following
where are constants. We see that the constant that relates and depends on the particular choice of the constants in the linear combination of exponentials. The most general solution of can have eight terms, each one with a proportional constant and the constant would depend on all these .
iv.4 Solution 4
This model has three bosonic components, one is constant, with a potential different from zero. These are
After imposing the only non-trivial condition and solving the equations we have a wave function solution with just one independent component, . The form of and its relation with the rest of the components is
and a constriction for this model arises, .
We have seen so far that these reduced lower dimensional models have exact solutions. We also found that for some of the configurations we have used, the arbitrary constants in the components of the wave function solutions can be chosen so that they are normalizable. We could make an extension of the method to a higher dimensional model but, as the dimension of the matrix representation for grows exponentially with the space dimensions, it would be hard to handle the complete 9-dimensional problem. It will be needed a powerful computational method to solve the complete matrix model, even for the SU() group and even harder for a larger group. In the next section, we solve a particular 4-dimensional model. Our intention is to show that the method can be directly extended to more dimensions and to any SU() group even though, as we mention, these extended models will be difficult to handle.
V SU(2) 4-dimensional extension
In the 4-dimensional extension for the SU(2) group there are 12 bosonic variables , and with the number of supercharges is 4. The number of fermionic variables is also 12 and the dimension for the matrix representation of these variables is . The representation we choose for the twelve matrices that satisfies the algebra eq. (4) is the following
where are the Pauli matrices. The Dirac representation for the matrices we use is
In this case is the identity matrix. Then, for example, the first supercharge takes the following form
In this extension we have a richer structure and we can search for more interesting potentials. We notice that the search for the ground state solution is much more complicated because we have to solve systems of 64 coupled partial differential equations for the 64 components of the wave function. In this extension we will also restrict ourselves to solve some reduced models with particular assumptions on the bosonic degrees of freedom.
Vi Solutions to the extended model
Using the matricial representation in eq. (22) we need to solve, again, systems of matricial partial differential equations. In this case, every supercharge and gauge symmetry operator is a operator matrix acting on a component wave function .
As an example we begin with the following model. We select two variables to be different from zero, , . In this particular model, we see from eq. (2) and eq. (23) that it is not necessary to impose the restrictions for any . We need then to solve, for example, As in the previous models the components of the wave function can be identified and one gets a solution with just two independent components , and . The independent components appear finally coupled in only two differential equations
These two coupled components are related to other six components of the wave function in the following manner
There are also other pairs of components coupled in the same way, here we write all these pairs of coupled components
If we identify all these pairs as one, then all the rest of the components are related to the only independent pair in a similar way as in eq. (26) and the wave function has only two independent components. This, now exact, reduced model obtained following our general procedure would correspond to the only two component one guessed in N3 () to analyze the spectrum of the Hamiltonian eq. (1). Our quantization approach leads to similar features. It allows however an exact solution which satisfies also the Hamiltonian operator including the fermionic sector at once. A continuum spectrum of the Hamiltonian is an expected feature in the connection between the supermembrane and D0-branes 1.1 (); N3 (). For several possible choices of the bosonic variables one gets trivial solutions. Here we show solutions to some non-trivial models.
vi.1 Solution 1
Another choice of a model with three bosonic components is the following
with all other bosonic variables equal to zero. In this case, we should impose the condition , with the consequence that many components of the wave function vanish. The only components that are different from zero are