THE HEISENBERG GROUP



|THE HEISENBERG GROUP |

Alberto Casal Grau (772510 P398)

e-mail: duendemoskeado@

SCHEME

• INTRODUCTION AND MOTIVATION

• NOTATION

• MATHEMATICAL TOOLS

• POISSON BRACKETS

• MORE DEFINITIONS

• HEISENBERG GROUP

• DEFINITION

• HEISENBERG ALGEBRA

• ADJOINT REPRESENTATION

• LIE BRACKETS

• SEMIDIRECT PRODUCT

• UNITARY REPRESENTATION

• CONCLUSION

• QUANTUM MECHANICS

• LITERATURE

INTRODUCTION AND MOTIVATION

The use of continuous groups and their unitary representations is not something new in theoretical physics, there are many examples of good and reasonable physical descriptions of nature and its development involving such algebraic structures. Quantum Mechanics show very reasonably how the two-valued representations of the rotation group are applied in the description of the spin of electrons; and Spherical Harmonics are indeed nothing more than irreducible representations of the same rotation group. As a matter of fact, if we wish to investigate the dynamics of a system described by classical Hamiltonian mechanics, we will need to learn more about Lie Algebras and Group Theory. Actually, there are very many examples in physics involving Lie Algebras, and even more; we could say these mathematical theories are nowadays essential to construct well-behaving fundamental structures and formulations for the most important theories in modern physics.

Symmetries and fundamental conservation laws are both the most important objects we deal with and we know that our interpretations and descriptions of nature cannot avoid them; at this time, the search for symmetries has taken us to learn, teach, study and apply many of these mathematical theories. Lie Groups, Lie Algebras and their Representations, as well as many other algebraic concepts and relations are not new, many of them were developed more than a hundred years ago, but they were only mathematical concepts, abstract concepts, just like strange tricking games mathematicians worked out to have fun and think hard. It has been in the last century, mainly due to the development of Quantum Mechanics and Particle Physics, when they have turned out to be the strongest and more powerful tool we may use to explain and describe such an interesting and complicated world we live in.

It has been very puzzling for physicists since 1950, and later on, the great success of such complicated mathematical structures explaining fundamental groups and interactions of tiny particles. But even more, it has been really amazing, because there have been also some new particles predicted by models and discovered afterwards on later experiments. It can be, of course, a bit dangerous, because we may get lost somewhere in an abstract huge strange superspace of algebraic methods in such a way that we feel completely unable to bring some knowledge about any feature, and even worse: we may suddenly realize we are unable to find the way out of such strange abstract world to finally conclude that what we are doing doesn’t make any sense at all! With respect to this, someone said once that symmetry is a tool that should be used to determine the underlying dynamics, which must explain later on the success (or failure) of some technique, but it can never be a substitute for physics itself. Mathematics and abstract theories are used as faithful tools to describe real phenomena in nature, but they can never be the final purpose.

As we have already said, Group Theory is now playing a central role in modern physics, and it is not strange. Many interesting facts are now supporting these Lie structures, like, for example Heisenberg´s isotopic spin, which can be understood pretty well in terms of a group of unitary matrices acting on protons and neutrons; and the discovery of the omega particle (() in 1964, which was predicted by SU(3) very precisely. This last one was a strong strike, and most of the possible remaining doubts about all this representation stuff and abstract theories were vanished, so they are now much more accepted for everybody. More recently, there has been a successful development of non Abelian gauge theories of strong and electroweak interactions.

But in this paper, we are not going to talk about all these great challenges of ultimate unification theories, gauge theories, the quark model, SU(5) or SO(10); we just want to recall another problem we’ve being fighting with since we met Hamiltonian Systems. These are very important because they offer a nice description for some motion equations of different kinds of observables. At the very beginning point, we find a wide set of smooth functions which may represent some vector fields or other objects in terms of appropriate operators of well known obsevables. We will start from an already known behavior of some kind of functions and, after that, we will be ready to introduce a more general way to focus and describe the whole problem, and furthermore, understand what happens and how.

We may be interested in dynamics once again, and we may think about the functions and the operators we use to describe them. The most general form of such functions is the polynomial one, so, we may think about simple constants, linear polynomials, quadratic polynomials, combinations of them, and many more. But let’s stop there, we don’t consider non-linearity, now. We are also used to deal with the fact that we can find and establish proper relations between physical observables and certain operators. We wish to relate univocly these operators to the functions in such way we can approach the solutions starting from different points separated apart but cleverly related together in a proper known way. And we certainly do this, although we know nothing about the results nor the physical measurements, yet. To fix ideas, think about the position and the momenta operators in classical mechanics, their associated smooth functions on the canonical variables, say, the kinetic terms; some external constant vector field, and the way they develop in time; all this together with some differential equations, leads us to Hamilton´s equations of motion for such a system, which are very nice and we may also solve. Now, we do know things can be more complicated than that, so, perhaps we wonder about the functions; say, could we solve problems for any function we can imagine? Since functions are mathematical objects, they can be much more complicated than just a linear or a quadratic polynomial, but we would hardly understand them if we don’t understand the simple ones first. In any event, we should also try to point our view to the ones with a correspondence with a physical real object or, at least, a good approximation. It turns out that actually polynomials can describe many physical situations, and are many times a pretty good approximation to reality. So, we realize this is a good approach, and then we may go ahead.

Of course, things certainly can be much more complicated, the world shows many variations and fluctuations all over it, and non-linearity is indeed a fact, it occurs. But we will stop before reaching non-linearity, and we are just going to look at the space of all polynomials up to second order, not more. If they are homogeneous or inhomogeneous is important because they may have different behavior. The first ones are easier to deal with, and it is known they obey some wellknown properties and relations which involve algebraic structures that work for very many cases. But we cannot use such structures when we handle inhomogeneous polynomials whenever we wish to, they simply don’t work the same way and cannot be applied. For this reason, we would like to find another algebraic structure of the same kind but different from the first one. There are always remaining things that we may keep (always hold), but there are also things which are very useful for some purposes but show nothing about others. Lie Groups and Lie Algebras have the property that they may be different from one another and have also different properties, but paradoxically, they are found to be lying over other related strong structures which may be understood, and therefore described in terms of the physical information encoded within the maths.

So, the motivation to look for Heisenberg Group, was to describe the behavior of the set of inhomogeneous quadratic polynomials in Hamiltonian Systems under some determined boundary conditions we will discuss later, using results and relations for the set of homogeneous quadratic polynomials. The main question could read: ”Is it possible to extend and apply the results for the algebra of Homogeneous Polynomials to the one of Inhomogeneous Polynomials?”.

NOTATION

• V = Vector Space

• X = Symplectic Manifold

• F(X) = Space of smooth functions

• D(X) = Lie Algebra

• Ham(X) = Space of Hamiltonian Vector Fields (subset of D(X))

• Sp(V) = Symplectic Group

• Mp(V) = Metaplectic Group

• L2(X) = Hilbert Space of Wavefunctions

• f , c = Functions

• M, RH, (, (, ( = Representations

• h(V) = Lie Algebra associated to Heisenberg Group (Heisenberg Algebra)

• H(V) = Heisenberg Group

• S = Unit circle in the Complex Plane

• ( = Symplectic form defined on a Vector Space

• ( = Vector Fields

• q, p = Canonical Variables

• q, p, v, u = Vectors

J, K = Groups

• z, z´ = Complex Numbers

X, Y, Z, a, b, c = Possible Group Elements

IQP = Inhomogeneous Quadratic Polynomials

HQP = Homogeneous Quadratic Polynomials

HLP = Homogeneous Linear Polynomials

ILP = Inhomogeneous Linear Polynomials

MATHEMATICAL TOOLS

POISSON BRACKETS (FIRST DEFINITION):

Let fj (with j an integer) be any functions belonging to F(X) on the 2n-variables qi, pi, with i = 1,2,…..,n. It is defined the Poisson Bracket of any two of them as the following:

[pic][pic]

Now, when we study how a physical system evolves with time we are used to deal with a function H (the Hamiltonian) which depends on several variables in a suitable basis (generalized coordinates) and gives us all the information we need in order to give a mathematical description of the situation. This dynamics are contained in some differential equations we must solve which are Hamilton´s equations. These are a consequence of some transformations made on the functions.

We are not interested now in the whole formulation of Hamilton´s theory, but we must remember that what makes it consistent are these convenient transformations. In the physics literature these transformations are the so-called ”infinitesimal canonical transformations”, while in a more general mathematical language they have the name of ”infinitesimal symplectic transformations”. As we want to state things from a general point of view, we will consider the second name and pretend we are just dealing with mathematical objects.

SYMPLECTIC FORM:

A symplectic form is a non-degenerate antisymmetric bilinear form which can only be defined on an even-dimensional vector space. Any tangent space to an n-dimensional symplectic manifold is endowed with such a symplectic form. Evolution can be described then by a continuous family of symplectic transformations on the phase space which preserve this form.

It is defined the Symplectic Form of an arbitrary n-dimensional Symplectic Manifold as the bilinear form of any two functions belonging to the set, as follows:

[pic] (( is the outer product defined on the manifold)

This form allows us to define a Vector Field in a convenient way, in terms of a smooth function on 2n-variables and its derivatives.

VECTOR FIELD:

[pic]

The trajectories of this field are the solution curves of a system of differential equations, which will provide the time evolution information of the variables:

[pic] [pic]

These are no more than Hamilton Equations in the physics literature (the function f characterizes the energy of some physical system, usually labeled by H).

[pic]

POISSON BRACKETS (SECOND DEFINITION):

[pic]

PROPERTIES:

Antisymmetry: [pic]

Derivation property: [pic]

Jacobi Identity: [pic]

MORE DEFINITIONS:

Homomorphism: It is a linear map which relates two sets (groups, vector spaces, etc.) preserving an operation defined on them in such way there is a one-to-one correspondence among their elements and its kernel is the empty set.

[pic] (φ is a map)

[pic], [pic].

(where “◦” reads for any possible composition or operation defined)

Isomorphism: If the preceding holds and at the same time the image of the map is precisely the final set, then such a map turns out to be an isomorphism. Then it is said to be invertible (g ≈ h).

Automorphism: It is a homomorphism from a certain set (again any group, vector space, algebra) onto itself (g→g).

Continuous one-parameter-subgroup: It is some linear function which operates on a group providing a continuous parameterization of the elements of the group with the parameter being a real number. The product of two of these group elements must be equal to the operator acting on the sum of the two elements, and it may only be defined in a small neighborhood of the identity element of the group. This unit element is usually taken to be zero, for simplicity. This subset is then the generator of an infinitesimal symmetry carried out by the group and it will be described by the derivative of the function evaluated on the unit element.

Symplectic Lie Group (Sp2nC): It reads for the group of automorphisms acting on some even-dimensional Complex Vector Space which preserve a non-degenerate skew-symmetric bilinear form defined for two elements of the space. Its Symplectic Lie Algebra would then be the group of maps from the space onto itself (endomorphisms) preserving now an antisymmetric bilinear form.

HEISENBERG GROUP

Let Ham(X) denote the space of all the vector fields on the form Ψf. There exits a linear map from F(X) onto Ham(X) sending each function f into the corresponding vector field. If X is connected, the kernel of this map will be the set of constant vector fields. We can express this by saying that the following one (D diagram) is an exact sequence of linear maps, that is, each map’s Image is the Kernel of the next,

[pic]

(D diagram)

Poisson Brackets can turn vector spaces into Lie Algebras, which are also vector spaces, but endowed with a binary operation between their elements which gives another element in the vector space. Poisson Brackets will be then labeled as Lie Brackets. This way, the maps lying on the above sequence are, in fact, homomorphisms among Lie Algebras. R is the algebra of Infinitesimal Translations and also the image of the space of HLP (which is not a subalgebra of F(X)), while the other two have already been labeled. As the space of HQP and IQP are smooth functions, they form two subalgebras of F(X). Now, let’s see what we know for the moment, that is:

▪ R2n is a commutative subalgebra of Ham(R2n),

▪ There exits a homomorphism sending a function from F(R2n) to a vector field of Ham(R2n) and has the constant fields as kernel,

▪ There exits an isomorphism between the algebra of Infinitesimal Translations and Ham(R2n),

▪ And another homomorphism relates the space of ILP and R2n.

We are looking for some subalgebra of F(X) able to project isomorphically onto (2n. Since for the moment we are only dealing with Lie Algebras, this kind of mapping is not possible to construct; and then we should look for some other suitable set which carries out with those properties. It turns out that such a group is the Heisenberg Group (HG from now on), which will allow us to see the problem from a wider point of view.

Let’s then consider, instead of the Lie Algebra of Infinitesimal Translations, the group of all translations (which is a Lie Group), also labeled the Group of all Finite Translations. We will see how it is possible to construct then a homomorphism between this group and HG, and also a nice relation with the space of IQP, as desired.

HEISENBERG GROUP:

Let V be a Symplectic Vector Space, ( its symplectic form and S the unit circle in the Complex plane. The Heisenberg Group is the set:

[pic]

endowed with this multiplication rule:

[pic] ( v1, v2 ( V and all z1, z2 ( S.

First of all, we check associativity among elements in the group, doing the following:

[pic]

[pic]

[pic]

[pic]

[pic]

=[pic].

Second, we look for the unit element, named (e1, e2):

[pic];

[pic];

[pic]

[pic]

( [pic], which is the unit element of the group.

Third, we calculate the inverse element for any other element of the group, say (v´, z´);

[pic]

[pic]

[pic]

And finally, its useful to state how conjugation is developed in the group, in order to define its associated Lie Algebra, the Adjoint Representation, and a Unitary Representation, as we will see.

[pic] (for conjugation)

HEISENBERG ALGEBRA:

It is defined the Lie Algebra associated to the group as the following vector space:

[pic]

ADJOINT REPRESENTATION:

The Adjoint Representation is given by taking the derivative at t = 0 of conjugation in the group, and provides a map from H(V) onto h(V). We take (tu, eit( ), which is a one-parameter subgroup of HG, and we get:

[pic]

=[pic]

[pic]

[pic]

[pic], evaluated at t = 0, gives:

[pic]

So, the Adjoint Representation is given by the formula:

[pic]

LIE BRACKETS:

According to the most general definition for Lie Brackets, that is, through the Adjoint Representation, we have:

[pic]

[pic]

Taking V = (2n, we can construct an isomorphism between Heisenberg Algebra and the algebra of ILP under Poisson Brackets. If n = 1 then V = (2; and the corresponding symplectic form would be:

[pic]

Then we can identify the vector (1 0) with the linear function that sends (q p) to the number (((1 0),(q p)), which comes out easily from the computation. Then the element ((1 0), 0) ( h(v) is identified with p, –q as the image of ((0 1), 0), and the constant function 1 from ((0 0), 0), which is the unit element of the algebra. This way, we have the relations:

[pic],

and (p, -q( = 1.

The first one holds for elements belonging to the algebra, and the second one holds for the space functions. This identities show an isomorphism between both sets. It comes now clear that we could do the same computing for the case V = (2n, and two vectors q, p of (n . The vectors turn out to be the generators of HG, and obey these important multiplication rules:

[pic]

To connect everything, we would like to find a representation for HG on the space of the wave functions L2((n), but in order to do so we need one more important definition.

SEMIDIRECT PRODUCT:

Let J and K be groups and J act as automorphisms of K. We denote the action of some X ( J on a ( K by Xa, and the group multiplication in J and in K by (. It turns out that the set of pairs J ( K, together with the multiplication law stated below, form also a group, and it is called the semidirect product of J and K.

Let X, Y, Z ( J and a, b, c ( K.

Multiplication law: [pic]

We check the properties achieved by the new group.

Associativity:

[pic]

[pic]

[pic]

Unit element, e=(e1,e2):

[pic]

[pic]

Inverse element:

[pic];

[pic]

[pic]

DEFINITION (REPRESENTATION OF THE SEMIDIRECT PRODUCT):

Suppose ρ and τ are representations of J, K on the same space V.

Suppose:

[pic]

We define a representation σ of the semidirect product of J and K by:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Proving that ( is a representation of the semidirect product, as we wanted.

UNITARY REPRESENTATION

We may now construct the Unitary Representation of the group on the space of wavefunctions according to the multiplication rules for group elements. We can make an optical approach, taking the following representations of generators this way:

[pic] (for translations)

[pic] (if we change the phase)

Now, in order to check if this is indeed a representation, we must see if the next two products give the same result,

(1) →[pic] (product of representations);

(2) →[pic] (representation of the product);

Where we have used the short notation for the generators of h(V) [pic]

Using the laws given for the group, we can get:

[pic]

[pic] for any integer n.

We took the value of RH (0, z) to be zn, in order to make the representation irreducible. Since (o, z) is in the center of HG, RH must be the multiplication by a scalar, and for RH to be a representation, this scalar must be zn for any integer n. Now, if we take n = -1, we see clearly that:

[pic].

We may now define the action of the representation on some element x this way:

[pic].

Let [pic]be vectors from HG, that is:

[pic] [pic]

We have just said that: [pic].

But now we can also compute the global action of the product of two such representations applied on some vector x ( (n and it comes out that:

[pic]

[pic]

It is important to notice that the action of RH is performed first on the element with the primes (which appears first), and afterwards on the one without primes (which appears second); because it seems it should be done just the opposite way round. Of course, if the proper changes are done, it would be possible to do the opposite way, obtaining the same final result. Anyway, some more operations lead to:

[pic]

[pic] [pic]

CONCLUSION:

The group Sp(V) preserves the linear and symplectic structure of V and acts as a group of automorphisms of HG.

The group Mp(V) (called Metaplectic Group) is the double cover of Sp(V) and therefore also acts as automorphisms of HG.

Now, let M be the metaplectic representation of Mp(V) and let’s also consider the Heisenberg representation RH on the space of wavefunctions. Using the definitions of the semidirect product and its representation and taking:

[pic]

we can write:

[pic]

It is possible to check that the Lie Algebra of the semidirect product in this case is the space of inhomogeneous polynomials. Since we have constructed the representation of the semidirect product, we have then constructed a “quantization procedure” for all inhomogeneous quadratic polynomials. It is also possible to show that RH is the unique irreducible representation of HG (theorem of Stone and von Neumann, Mackey 1968, pp. 49-53), with the property that [pic], where I is the identity operator. RH determines a unique projective representation of the symplectic group from general principles of representation theory, and hence a unique method of quantizing quadratic polynomials.

QUANTUM MECHANICS

Without giving any further details, this is a short mention of some other developed applications involving these sets and relations. Heisenberg´ contributions have been decisive to build a huge part of QM theory and as everyone should know, his Uncertainty Relations provide a very clever and indeed fundamental description of the world.

In relativistic field theory of free bosons, the bosonic hermitian fields and the canonical momenta used to describe the dynamics again can be related by the Lie Brackets defined for Heisenberg Algebra for the case of a single boson confined inside the unit circle in the complex plane and with an infinite number of possible states.

The idea of confining the field inside the unit circle simplifies very much the problem and makes it solvable. In fact, the same fields (which are operators) can be expressed in terms of the generators (also operators) of the algebra. These operators are basically the same ones, upon some proper constant rescaling, that appear in the quantum description of the Harmonic Oscillator. Actually, its quantization has much to do with this Quantization Procedure for Inhomogeneous Quadratic Polynomials we have analyzed along this paper.

LITERATURE

“Symplectic Techniques in Physics”. Victor Guillemin, Shlomo Sternberg. Cambridge University Press.

“Lie Algebras in Particle Physics (From Isospin to Unified Theories)”. Howard Georgi. Lyman Laboratory of Physics, Harvard University. Cambridge, Massachusetts, 1982.

“Symmetries, Lie Algebras and Representations”. Fuchs and Schweigert. Cambridge University Press, 1997.

“Representation Theory (A First Course)”. William Fulton, Joe Harris. Springer-Verlag, 1991.

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