Understanding Quantum Theory

Understanding Quantum Theory (I)

Y. C. Leung

Department of Physics

University of Massachusetts Dartmouth

N. Dartmouth, MA 02747

A pedagogic introduction of quantum theory is presented. It attempts to answer some of the nagging questions confronting students learning the theory for the first time, and explains the fundamentals of quantum theory from a perspective not commonly found in textbooks, with the intention of stimulating discussions in teaching quantum mechanics.

I. INTRODUCTION

Every year thousands of students are being introduced to a course in quantum mechanics for the first time. Having barely overcome the operational skills demanded for classical mechanics they are then asked to accept a totally different skill in order to deal with quantum mechanics. The transition from classical mechanics to quantum mechanics is certainly not easy for most of us. Some people compare it to advance from geometric optics to wave optics, but I don't think the comparison is well made for it underestimates the conceptual difficulty. The difference in technique between classical mechanics and quantum mechanics are due to major conceptual differences and not just manipulative needs. For one thing, the complex number i appears everywhere in quantum mechanics. Sure, we have seen complex functions employed in electrical circuit analysis and in wave optics, but they were introduced for convenience, as operation with real function is perfectly feasible, not because they are indispensable as in quantum mechanics. Such seemingly minor issues are baffling to an inquisitive mind, and if not explained (as no instructor of mine ever tried to explain the ubiquitous i ) can be stymieing to a student's progress.

When I first studied quantum mechanics, I was mystified by many of the highly specialized mathematical operations which I was instructed to follow. By the widest stretch of imagination I could not conceive how physicists like Schrodinger, Heisenberg, and Dirac could come up with ideas exemplified by relations bearing their names. It took me many years after my first course in quantum mechanics to be somewhat comfortable with its ideas. I would like to share my understanding of quantum theory, the logical path of development it follows, with all students of quantum mechanics for an exchange of views.

Let me apologize that I shall not give historical references to the ideas discussed here since I have not done a complete study of the history of quantum theory. This is to avoid misrepresentations.

II. THE "GAPING-HOLE" PROBLEM

We shall construct a quantum theory starting with the recognition that the physical values of angular momentum come in integral multiples of some basic unit Ћ which is Planck's constant divided by 2π. Since angular momentum is represented by a three ­dimensional vector, we should qualify the above statement to mean that any component of the angular momentum, if measured, comes in integral multiples of Ћ. This property is known as quantization of the angular momentum, and is generally accepted to be firmly grounded.

In classical physics all components of an angular momentum may take any real value, and quantization is contrary to the basic tenor of classical physics. Such a direct confrontation with classical physics must result in a reformulation of its expression, and so let us begin with angular­momentum quantization and see how it can be accommodated in a properly formulated theory.

In classical physics, the angular momentum L of an object about a point is given by the cross product of the vectorial distance r between the point and the object, with p, its linear momentum:

Both r and p are three­dimensional vectors. Since the components of r and p can take on arbitrary real values, the components of L must necessarily take on continuous real values. Such a definition of L is thus incompatible with the requirement of angular­momentum quantization, for starting out with continuously allowable magnitudes of r and p, (1) must necessarily yield continuously varying values of L. How then should the mathematical structure of (1) be modified so that the quantization of L would come about? (Bear in mind that the dimension of angular momentum is still the product of those of r and p.) How is it that L will miss all the disallowed values but hit only the allowed values? The disallowed values appear like "gaping holes" on the real number scale. We shall look at the "gaping­hole" problem as a first step in solving the quantization problem.

We are thus forced to explore different mathematical schemes. For instance, instead of employing the real­number field as the basis of computation, we can try others, like the complex­number field, the quaternion field, or even the octonion field, etc, to describe the physical parameters. In other words, representing the components of r and p by complex numbers, for example, and see if the resulting product as expressed in (1), or taking just the real part of that product, would give rise to "gaping holes." Let us examine a particular case: let r be perpendicular to p and their lengths be represented by complex quantities reiφand peiθ, respectively, where r, p, φ and θ are all real numbers, then the real part of L is given by:

It does seem then that there is a freedom in adjusting (φ + θ) so that the final result falls on a certain discrete value. However, if we were to accept this scheme, we must resolve the problem of assigning the phase angle of a complex number to each physical quantity. As to be expected, it is not likely that this scheme will ever solve the "gaping-­hole" problem in a natural way. This is because r and p in (1) are connected by a product, which satisfies the distributive law, such that:

meaning that L would change infinitesimally by dL, if r were changed infinitesimally by dr. Thus, if either r or p is a continuous variable, L must be a continuous variable by (3). It is not likely that by employing complex numbers would change this picture, and thus complex numbers would not solve the "gaping­hole" problem.

Quaternions are numbers which are non­commutative, and octonions are numbers which are non­associative as well as non­commutative. Again it is not likely that by rewriting the components of r and p in terms of these numbers, but retaining (1), will solve our problem. We must look elsewhere for a solution.

Actually the "gaping-hole" phenomenon is most glaringly exhibited in the diagonalization of matrices. When a square matrix is diagonalized, there are just so many eigenvalues associated with the matrix. The eigenvalues do not form a continuous set, even if the matrix consists of continuous elements and is infinite dimensional. Thus, the eigenvalue approach could be a prime candidate for solving the "gaping­hole" problem. Indeed, as we all know, one of the earliest formulations of the quantum mechanics, attributed to Heisenberg, is based on the representation of mechanical quantities by matrices and their experimentally observed values associated with the eigenvalues of the appropriate matrices.

Heisenberg's construction of the matrix mechanics is based on certain commutation relations between the components of r and p, which mystified me. Such relations formed the cornerstones for the structure of quantum mechanics however. The fact that I could not understand their origin, even with hindsights, retarded my progress in working with quantum mechanics. The commutation relations involve Ћ and i. I had an easier time in accepting Ћ  (if only for dimensional reason), but I was baffled by the appearance of i. I feel that many students may encounter the same obstacles as I did in learning quantum mechanics. For this and other reasons I try to present here an alternative way to introduce the concepts of representing dynamical variables as matrices and to demonstrate the emergence of such commutation relations as a natural consequence.

III. MAPPING TO REAL NUMBERS

When we open a mathematics book on matrices, we see that matrices are introduced under the heading of linear algebra, or sometimes, linear operators. They form prime examples of a linear algebraic system which is not too trivial. Whether or not linearity is crucial to quantum mechanics is a very profound question. We shall not analyze this issue here, but use linear algebra as a guide for our exposition.

For a real symmetric matrix all of its eigenvalues are real numbers, which remain unchanged even if the matrix takes on a different look after a similarity transformation. The eigenvalues of a matrix therefore represent certain characteristic properties of the matrix, expressible by a set of real numbers. Finding the eigenvalues of a matrix is synonymous with the diagonalization of the matrix, and we shall use these two expressions interchangeably. We may accept diagonalization by its most primitive function which is a mapping of the matrix onto the real numbers. The availability of this function, though no cause for celebration, is crucial to the mathematical description of physical phenomena which are only expressible in real values. Let us review this mapping with illustrations to highlight its significance. Consider the following situations:

1. In electrical circuit analysis, physical quantities like current and voltage are expressed by complex functions and their actual magnitudes are given by the real part of these functions; thus when we are dealing with complex numbers, z, the following operations are commonly employed: |z|, Re {z}, Im {z}, which assign mappings of z to the real numbers.

2. In dealing with real vectors A, B, etc, the mapping is accomplished through the scalar product A.B (more generally, contravariant vectors are mapped to the real numbers by the corresponding covariant vectors via an inner product). The components of each vector may take on different values in different rotated frames of reference, but the scalar product is unaffected by such vector transformations. Physical result expressed by the scalar product exhibits thus an invariance under rotation.

3. A real vector may be represented by a column matrix with each component of the vector occupying a position in the column. Call this a contravariant vector, while the covariant vector is represented by a row matrix of the same size. The scalar product of these vectors is given by the multiplication of these matrices (row matrix to the left of column matrix) following the usual matrix­multiplication rules:.

1. The eigenvector vi (represented by a column matrix) of a square matrix M accomplishes the following result:

• 1. where the superscript T denotes a transpose which converts a column matrix into a row matrix, and λi the eigenvalue. If matrix M is symmetric (and non­singular), the eigenvalues λi will be real. Here we see the eigenvector vi accomplishes the mapping of the matrix M to a real number which is its corresponding eigenvalue. This process is equivalent to the diagonalization procedure with the diagonal elements composing of all the eigenvalues. The diagonalization procedure represents a mapping of the matrix M to a set of real numbers.

1. In linear algebra, matrix M plays the role of an operator which transforms a vector into another:

• 1. where u and w are column matrices. In this capacity M is similar to other linear operators such as certain (linear) differential operators which transform one function into another. The concept of eigenvectors can be generalized to eigenfunctions, so that a column matrix is represented by a real­valued function fi(x) of continuous variable x, and (5) by

• 1. where O is a linear differential operator taking the place of matrix M, and matrix multiplication is replaced by integration. This expression accomplishes a mapping of an operator O to a real­valued li. Thus linear differential operators and matrices behave similarly under this operation.

1. Taking the trace of a matrix provides also a mapping of the matrix to the real number, which like the matrix's eigenvalues is unchanged when the matrix is transformed under a similarity transformation. It reflects clearly an inherent property of the matrix, though it reveals less about the matrix than the eigenvalues. This operation appears less frequently in quantum mechanics when the kinematical properties of a single particle state is sought, but is employed extensively in statistical mechanics where one is interested only in the ensemble averages. Others, like the determinant of a matrix, etc., are available operations under consideration.

These examples illustrate the various forms that this mapping takes, some of which are more familiar than the others, but they all play the role of converting a complex mathematical form to a real number, which may then be associated with the value of a physical measurement. Employing this mapping frees us from working entirely in the real­number system as in classical mechanics, allowing us to contemplate on working with more abstract mathematical operations knowing that whenever we need to confront analysis with experimental observations a mapping to the real number is available.

If a matrix M is not symmetric, some of its eigenvalues may be complex numbers and thus fall outside the real­number system. Working only with symmetric matrices is an unnecessary limitation to impose on oneself. On the other hand, the eigenvalues of all (non­singular) complex matrices are expressible as complex numbers, and the analysis is closed within the complex­number system. Thus, the complex­number system is always employed. The eigenvalues of a complex matrix M will be real if M is Hermitian. The diagonalization of a Hermitian M achieves a mapping of M to the real numbers. Hermitian matrices need not be symmetric, freeing us from working entirely with real­symmetric matrices.

We shall however always initiate our discussion on the basis of the real numbers, and bring in the complex numbers only when there are compelling reasons to do so.

IV. GENERATOR AS DYNAMICAL VARIABLE

Returning to our original problem of angular­momentum quantization, we shall now explore the possibility of representing angular momentum as a matrix or some linear differential operator for which a mapping to the real numbers procedure is available. The crucial question is clearly: How should it be represented? Well, where do we start? We have seen above that matrices transforms one column vector into another, and differential operators transform one function to another. To gain some insight into the problem, let us first examine how mathematical functions transform.

Let us begin with simple examples. Consider first a continuous function f(x) of the three­dimensional spatial variable x. When this function is displaced by an amount a to f(x+a), called a translation by a, it can be accomplished by applying a translational operator T, as follows:

To find out what T is, we only need to expand f(x+a) into a Taylor series:

Thus, T(a) = exp{a.∇}, with the exponential function defined by the series preceding f(x) in (9). Expressed in the exponential form, the content of the operator T(a) is clearly displayed. The argument of the exponential function consists of the dot product of a and ∇, where a is just a parameter claiming the amount of translation desired, and the remaining part, ∇, is called the generator. So far, there is no reason to introduce complex numbers and we shall stay with real functions and real variables for now.

In quantum mechanics, we are told that the momentum variable must be replaced by

which is a puzzling requirement. Supposedly, this replacement is demanded by the commutation relations of its components:

etc, an equally puzzling requirement. Not only is the commutation relation an unfamiliar concept, a complex number i appears out of nowhere. I think most students of quantum mechanics must have a great deal of difficulty in accepting such a prescription. I shall attempt to shed some light on it origin.

Noting that the generator of a translation is given by ∇, it is tantalizing to postulate that the linear momentum p, a dynamical variable, is to be identified with an operator in quantum mechanics given by Ћ ×  the generator of translation. (A quantity like Ћ  is needed here for dimensional reason.) The difference between this identification and (10) is just the complex number i. Why should there be an i ? Can't we not formulate quantum mechanics with only real variables? To gain further insight, let us look at another example.

If the suggested postulate is correct, then the angular momentum operator should be identified with the generator of a rotation. This association is intuitive based on geometric reasoning. Examine now the rotation of a simple three­dimensional vector x. Let the coordinates of the vector be x = (x, y, z) and the rotation is about the z­axis of the coordinate system by an angle

. The coordinates of the rotated vector will be x' = (x', y', z'), where

Let us write x' = R(

) x, where R() is the rotational operator. Such a relation takes on a concrete form if x and x' are represented by column matrices as follows:

The rotational operator R(

), represented by a 33 matrix, can again be written as an exponential function,

where

(14) can be checked by writing out the exponential in a series form as in (9) and making use of the fact that

and the series expansion:

Pursuing the identification suggested for p, Jz =

Lz is now the angular momentum operator.

The observed values of the angular momentum are contained in its eigenvalues. The eigenvalues of Jz can easily be found, and they are: +i

, 0, ­i. The eigenvalues are not real, as to be expected, since Jz is antisymmetric. Regrettably, we must conclude that the above identification of dynamical variable with generator is incorrect.

In the case of rotation, it is easily seen that, working with complex numbers and complex functions, Lz is make into a Hermitian matrix if a factor i is factored from it:

Lz' is now Hermitian, and its eigenvalues are , 0, ­, which are the components of a "unit" angular momentum. This suggests that the postulate can be made to work with complex matrices, by re­defining the generator with an complex number i in the exponent:

Similarly

In this manner, with the generator of translation given by 1/i

, the commonly known momentum operator p is indeed just { generator of translation}. The momentum operator p thus defined is Hermitian. Let us examine this point. Working with real­valued functions instead of matrices, an Hermitian operator O possesses the property that

The operator d/dx clearly is not Hermitian, since integration by part shows

where, in deriving this result, we have assumed that both f(x) and g(x) vanish at the limits of integration. The result shows that d/dx is in fact anti­Hermitian. Changing over now to work with complex functions, the scalar product is modified to be:

where g(x) receives a complex conjugation, indicated by *. With ­i d/dx as the operator, an integration by part on the scalar product yields:

showing that it is indeed an Hermitian operator, and its eigenvalue will be real.

I hope the above discussion will help to enlighten the reader on the reason why quantum mechanics adpots complex­number field for its analysis. It manages to achieve certain basic requirements, such as Hermiticity, conveniently. Not that the same analysis cannot be performed with real variables, but the operations would be more elaborate. As noted by Stueckelberg [1], quantum mechanics may be formulated in the real­number field on a real Hilbert space by introducing a certain anti­symmetric operator [I] which commutes with all operators serving basically the role of the imaginary number i. Indeed, the factor i appearing in the definition of a generator should more properly be regarded as a special anti­Hermitian operator rather than just an ordinary complex number (as in the sense employed for the study of the analytic properties of the scattering amplitude, or the group structure of the complex Lorentz group).

The momentum operator px obtained this way agrees with that derived from the commutation relation in (11), where both px and x are to be considered as operators. The appearance of the imaginary number i in the commutation relation is now clear. Since both px and x are Hermitian operators, their commutator must necessarily be anti­Hermitian. The imaginary number i on the right­hand­side of the expression is needed to achieve the anti­Hermiticity condition. Again, the factor i appearing here should more properly be regarded as a realization of an anti­Hermitian operator rather than just a complex number.

The postulate in associating the dynamical variables with the generators of their corresponding space­time transformations is a very profound concept. It replaces Heisenberg's commutation postulate. We shall elevate its status and call it the Dynamical Principle. In addition to translations and rotations, we can consider also Lorentz transformations for which there will be associated operators called boosts. Discrete transformations are also possible, like parity and time reversal, for which there are no classical counterparts since no generators exist for these transformations. The Dynamical Principle makes clear furthermore the connection between space­time symmetry and mechanical conservation laws. We shall discuss this at a later time.

Let us return to the rotational operator R(f). In general, we require R(f) to act on f(x,y,z), a function over the 3­dimensional space:

where x, y, z transform according to (12). To find R(

) we first consider an infinitesimal rotation with parameter , so that to the first order in

Also to first order in

,

A rotation by a finite angle

can be achieved by infinitely­repeated infinitesimal transformations as follows:

is replaced by /n in the limit n ­> , and the last equality is obtained by comparing the binomial expansion of the middle term to the exponential expansion of the last term. The generator of this transformation gives indeed the orbital angular momentum operator in quantum mechanics by the Dynamical Principle.

The last example shows that the generators can be found by just considering the infinitesimal transformations. For an infinitesimal Lorentz transformation along the +x­direction, to first order in

= v/c:

The generator for this transformation can be written down from inspection:

The other components can similarly be written down.

For discrete transformations, such as x ­> ­x, etc, there will be no corresponding generators as there are no classical counterparts. Following the introduced concept, each transformation operator can still be associated with a kinematical variable. Thus new entities, like parity, are admitted in quantum mechanics, and it is even possible to introduce new experimentally­verifiable ideas like the conservation of parity.

Other issues of quantum theory will be examined in a sequel to this article, where a purely geometric formulation of quantum dynamics will be proposed.

Reference

1. E.C.G. Stueckelberg, Helv. Phys. Acta, 33, 727 (1960).

Understanding Quantum Theory (II)

Y. C. Leung

Department of Physics

University of Massachusetts Dartmouth

N. Dartmouth, MA 02747

This is a continuation of a previous article by the same name with further discussion of issues related to quantum theory. The topics presented here, however, are intended for more advanced students of the quantum theory.

V. HILBERT SPACE

Let us review once more the type of relations that we need. First of all, we need to represent dynamical variables by generators which are expressible as matrices or linear differential operators, and then we need a mapping of these operators to the real numbers. We find that we can achieve this by solving the eigenvalue problem, which further suggests that it is preferrable to work with complex numbers and the complex vector space (since not all eigenvalues of real matrices are expressible as real numbers but all eigenvalues of complex matrices are expressible as complex numbers).

Our exercise with the angular momentum operator in either the matrix or differential­operator form shows that its eigenfuctions constitute a complex linear vector space and the operation leading to the eigenvalues is precisely that of taking an inner product. The inner product defined as a bilinear functional in this vector space provides a mapping to the complex numbers. Furthermore, the inner­product space may be taken to be the Hilbert space, and thus endowing the mathematics of quantum mechanics with the needed precision. Ever since the early days of quantum mechanics, the mathematical foundation of quantum mechanics had been assumed to rest on analysis in Hilbert space. Thus, the vector space we shall work with consists of functions which are normalizable (square­integrable) and expandable in terms of a set of basis functions (completeness), and operators are linear transformations within the vector space (automorphisms). We mention the last point here because it will play a role in the next section.

As long as the eigenvalues of the operator form a denumerable set, the Hilbert space provides a rigorous formulation of quantum mechanics, though in the case of continuous eigenvalues certain amount of "rigging" is needed [1]. Nonetheless, few physicists felt compelled to search for something new. The simplicity and definiteness of Hilbert space serve quantum physics well, at least as a starting point for all possible modifications.

The linearity of the vector space gives rise to the superposition principle in physical reality, which is the basis of wave­particle duality introduced in nearly all elementary quantum mechanics textbooks as a justification for the Schrodinger equation, though I personally did not find wave analogy very helpful as a concept.

Physical variables expressible as Hermitian operators within a commuting set of such operators can be sought as eigenvalues of the vector space. Eigenvalues of Hermitian operators are real, but in general an inner product may be complex and physical interpretation of an inner product, as in a transition process, is possible only by taking another mapping, its absolute square. For example, the scattering cross section is related to the absolute square of an inner product.

Thus the adoption of a complex Hilbert space injects new terminology like scattering amplitude and probability amplitude into the language. Such terms are unknown previously in natural philosophy. Are they truly fundamental, or just by­products of our choice of a complex Hilbert space? I have no good answer to this, and the question remains an enigma in my mind.

Consider however a Banach space which is a normed vector space, with similar properties as Hilbert space (but with a norm replacing the inner product). Since every inner product can be expressed as a sum of properly defined norms [2], it seems to me that either the Hilbert space or the Banach space is adequate for the purpose of quantum mechanics. Had the pioneers of quantum mechanics adopted the Banach space instead of Hilbert space to describe quantum mechanics, wouldn't we be employing a totally different vocabulary to describe the same physical results?

Admittedly, the Banach space is no more "natural" than the Hilbert space, and if anything, probably less so, but shouldn't the choice of space be made objectively? If the need for Hilbert space as a tool is only sufficient, but not necessary, shouldn't we treat the probability­amplitude type of interpretation with reservation: just operationally adequate but not conceptually inevitable? A clarification of this issue is much advised for all future quantum mechanics textbooks.

Actually most physicists are taking the role of an amplitude much more seriously than it is presented here. By imposing, for example, Poincare invariance on the scattering amplitude, conservation laws similar to those derivable from Noether's theorem are obtained. These results highlight beautifully the connection between space­time symmetry and kinematical conservation laws. On the other hand, working on the level of cross section many of these results will no longer be apparent. Are these then hints that the Hilbert space approach is truly necessary? I don't know if anybody has a definitive answer to this question, but will some clear­minded person tell us what logical steps or experimental verifications are needed to establish the necessity of Hilbert space in quantum mechanics?

VI. ELECTROMAGNETISM

Return now to the question of interaction in quantum mechanics. We have seen in our earlier discussion that we are led to the same representation of the momentum operator in quantum mechanics by either identifying it as the generator of translation, or following Heisenberg's commutation relations given in (11) of Paper I. We shall refer to these two methods as the generator approach and the commutator approach, respectively. Though the generator approach makes explicit the geometric underpining of kinematical operators like momentum and angular momentum, the commutator approach is more abstract and Heisenberg's commutation relations have been generalized to include not only the spatial coordinates but also the generalized coordinates. These are the canonical commutation relations which prescribe the all­important dynamical interaction in a quantum mechanical system. In order for the generator approach to be acceptable as a useful quantum mechanical concept it must duplicate this feat. Let us illustrate what we mean by this with an example.

In classical mechanics it is possible to describe a dynamical system by a classical Lagrangian L(qj,

j) which is a function of the generalized coordinates qj and their time derivatives

j, where j enumerates all such independent coordinates. A set of generalized momenta are defined through

Quantization of the classical theory proceeds via the canonical commutation relations:

For example, the classical Lagrangian of a non­relativisitic particle of charge e interacting with the electromagnetic field is given by:

where v is the particle velocity, c the speed of light,

and A are the electromagnetic scalar and vector potentials, respectively. The classical generalized momentum is given by:

The classical Hamiltonian is no longer a function of v and the following substitution is to be made:

with the result that:

The quantum mechanical Hamiltonian is obtained by replacing the generalized momentum p with , and is given by

where V = e

by . (We shall ignore the scalar potential for now, which is naturally included in a relativistic formulation.) The latter quantity is reminiscence of the definition of a covariant derivative which generates a parallel displacement of a vector over a curved spatial region. In that case, the quantity in the position of A is called the connection, which prescribes the curvature, or the geometry, of the region. A geometric interpretation of the electromagnetic interaction suggests itself by pursuing this analogy; however, the quantity in the position of A cannot be the connection in the usual sense of curved space because of the presence of the complex number i. Therefore, in quantum mechanics, it is the geometry of the space of states, and not the spatial region, that is of relevance here, as we explain below.

A highly instructive approach to the understanding of electromagnetism is to study the effect of a local gauge transformation on the theory [3], which proceeds along the line of reasoning given below. Since all quantum­mechanical amplitudes (scalar products in Hilbert space) are unchanged when its states are multiplied by a phase factor,

where

is a wave function and o is a real constant, quantum theory is said to be invariant under a constant phase transformation. What if o becomes spatially dependent (x)? The latter gives rise to a local phase transformation as the phase factor depends on each locality,

Would the theory be still invariant under a local phase transformation? Clearly all derivatives of the transformed states would not be,

which is not the phase transform of the original quantity, and thus the derivative operator

are consistently replaced by , called the gauge covariant derivative, invariance under local phase transformation is achieved if A would accompany the transformation with a corresponding gauge transformation:

. By a direct computation the following result can be shown:

which is the phase transform of the original quantity. The replacement of a derivative by a gauge covariant derivative is precisely what is being called for in (37) when electromagnetic interaction is present. Thus, quantum theory with electromagnetic interaction possesses invariance under local phase transformation, also called local gauge invariance. It is therefore argued that the appearance of the electromagnetic potential A, together with its gauge transformations, is necessitated by the possibility of a local phase transformation of its states. Such an explanation is not available in classical electromagnetic theory, and the meaning of gauge transformation remains inexplicable. Yang and Mills [3] extended this idea to encompass phase transformations which possess internal symmetry leading to the now­famous Yang-Mills theory, a cornerstone in modern quantum theory.

The geometric aspects of a local gauge invariant theory is succinctly luminated by viewing it as a mathematical fiber bundle theory which is a purely geometric construct. To illustrate its salient feature consider the following demonstration.

Let

(x) be the wave function describing the state of a single particle. Normally we use a single to describe the state for the entire spatial region. To study the local properties of

under a phase transformation we allow the spatial region to be covered by overlapping spatial patches, which may be arbitrarily small or infinitesimal. The electromagnetic potential A in different patches will be gauged differently, since it cannot be directly observed, and states of different fibers would belong to Hamiltonians of differently gauged A. It is claimed that electromagnetism is naturally included in the discussion if states belonging to different fibers but at the same spatial points are related by a phase transformation (see 42 below).

Let a and b denote two separate patches which share a common overlap, and the electromagnetic potential A specified over these two patches be denoted by Aa and Ab, respectively. Due to gauge ambiguity they may take on different values at the same point x in the overlap, but the difference is no more than the gradient of a function L (as they must yield the same magnetic field B =

A, which is an observable quantity):

Let

a and b represent respectively the wave functions of the system in these patches. At x in the overlap, they are required to be related by:

where S is a phase factor given by:

'a = a and 'b = b; with S given by (43) it can easily be seen that 'a S 'b. This shows that is not an admissible operator since it transforms states into new states not of the original space. Consider next,

which shows that , the gauge­covariant derivative incorporating electromagnetism, is indeed the proper operator on this space. Thus, electromagnetic effects would have been included if dynamical variables are constructed as appropriate operators on this space.

In the language of the fiber bundle space [5], the spatial domain provides the base space which is to be covered by overlapping patches. Within each patch all states generated by phase transformations constitute a fiber, which consists of therefore two components, the "horizontal" component denoting the spatial dimenisons, and the "vertical" component denoting the phase­related states at the same spatial point. The S factor in (43), called a transition function, satisfying certain compatibility conditions, is not limited to a phase factor, but must be some element of a group. The totality of fibers together with the transition functions among them comprises a fiber bundle. In the present case, transfomations along the "vertical" direction of a fiber form a U(1) group, and the transition functions also form a U(1) group, and the fiber bundle is called a U(1) principal bundle. In a general Yang-Mills theory the U(1) group may be replaced by any Lie group, and it is this group structure which specifies the nature of interaction.

We investigate gauge­covariant derivatives through the tangent space of the fiber bundle. Tangents to the fiber (i.e., tangents to the U(1) group of phases) constitute the vertical subspace of the tangent space, and the remaining subspace is the horizontal subspace, whose basis is provided by the gauge­covariant derivatives.

How natural is it to identify dynamical variables with the coordinates of the horizontal tangent subspace of a principal fiber bundle? I try to make a case for it in the early discussion by calling attention to the concept of generators, which form the basis for the tangent space over the space of states. Its generalization to a fiber­bundle space may still need further preparation in our mathematical background before the process appears natural. Such a preparation may well worth our effort, and it is being advocated here.

I feel that the geometric approach will eventually find its rightful place in the development of quantum mechanics. Its appeal, to me, rests largely on its minimal dependence on the classical formalism. Canonical quantization begins with the construction of a "classical" Lagrangian which is really a pure artifact since subnucleic particles like quarks do not exist in the classical domain. A quantum theory formulated without reliance on such an artifact is surely desirable, especially when one seeks to make sense of its classical limit. The mathematics of fiber bundles provides a enticing opportunity for us to achieve this goal, and efforts should be devoted to examine its possibilities and limitations.

Admittedly, not all interactions available to canonical quantization can be handled by the fiber bundle approach, but since all known fundamental interactions seems to be of the gauge type and fall within the fiber bundle description, there is even more reasons to search for its subtlety.

This article has unavoidably many omissions and shortcomings, but I hope it will generate further discussions in the teaching of quantum mechanics. I wish to thank my colleagues and students for many challenging discussions throughout my years of teaching.

References

1. For a discussion of the Rigged Hilbert Space, see for example, A. Bohm, Quantum Mechanics: Foundations and Applications, Springer­Verlag, New York (1979).

2. The following results can be shown to be true:

1. and

1. where (f,y) defines an inner product in a complex vector space with vectors denoted by f and y, and || f || = (f,f) defines the norm. See, P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity (second edition) Chelsea Pub. Co., New York (1957).

2. C.N. Yang and R.L. Mills, Phys. Rev. 96, 191 (1954)

3. We follow here a discussion given by Chen Ning Yang in "Magnetic Monopoles, Fiber Bundles, and Gauge Fields," Annals of New York Academy of Sciences 294, 86 (1977).

4. A readable text on fiber bundle space for outsiders is given by C. Nash and S. Sen, Topology and Geometry for Physicists, Academic Press, London (1983).