_{1}

^{*}

We demonstrate two points: 1) the formalism of quantum mechanics can be understood simply as a structure for the expression of the physical notion that not all observables can have values simultaneously; 2) the specific uncertainty relations can be derived (rigorously) by combination of the invariance principle with a general uncertainty relation based only on the existence of unspecified pairs of conjugate observables. For this purpose, we present a formulation of quantum mechanics based strictly on the invariance principle and a “weak” statement of the uncertainty principle that asserts only the existence of incompatible (conjugate) observables without specifying which observables are incompatible. We go on to show that the invariance principle can be used to develop the equations of motion of the theory, including the Klein-Gordon and Schrodinger equations.

While the famous uncertainty relation

Here, by the principle of uncertainty is meant simply the notion that there are sets of incompatible observables that cannot have precise values simultaneously. More particularly, the principle expresses the fact that there exist certain pairs of “conjugate observables”, a and b, the simultaneous values of which have irreducible intrinsic uncertainties Da and Db that satisfy an inequality of the form

where

Although the manner in which quantum theory is developed is hallowed by tradition, and readers steeped in a view of the subject they have come to terms with may not be open to an alternative interpretation, it is the contention here that the development outlined below simplifies the interpretation of the theory, and therefore deserves a place in the literature. Specifically, it addresses the first question that a student of quantum mechanics should have, namely, the question why the formalism of the theory must be changed from the formalism of classical physics in which observables are represented by mathematical variables. The historical approach to quantum mechanics provides a somewhat complicated answer to this question, whereas it is made evident here that the simple answer is the existence of pairs of observables that cannot have precise values simultaneously. Although the importance of the principles of uncertainty and invariance in quantum theory is well recognized, we are unaware of either a treatise on the subject that develops the theory strictly on the basis of these two principles or a place in the literature where the two points in the abstract are stated. Instead, the theory is either presented following the line of its historical development on the basis of the asserted wave nature of matter [

In the traditional developments of the quantum theory, the uncertainty principle is sometimes derived as a consequence of what are made to appear more basic or at least equally basic notions, and the fundamental nature of this principle is therefore obscured. In contrast, it is contended here that the uncertainty principle by itself is sufficient to both justify the entire mathematical formalism underlying the theory and to allow for a derivation of the quantum mechanical equations that result from it. Moreover, it is shown here that the development of the theory based on the uncertainty principle makes evident the fact that the quantum mechanical formalism proves useful only because it allows for a simple quantitative expression of the notion of incompatible observables. It is significant that the few rules and definitions used to define the formalism are no more numerous than those assumed in any formulation of the theory. That the resulting formalism is self consistent and leads to no physical content other than that embedded into it by the notions of uncertainty and invariance is demonstrated in detail in [

To make the points above, certain well known equations are restated in Section II and III.A for the purpose of showing that the content of these equations is that contained in Equation (1). Since it is understood that content is added to the theory through the invariance principle, Section III.B is intended to emphasize exactly how its content enters the formalism and to demonstrate that the principle leads to the existence of explicit pairs of conjugate observables. Specifically, it is shown to be a consequence of Equation (1) and the definition of the generator of an infinitesimal displacement, that the observable associated with the generator of the displacement is conjugate to a coordinate observable and has all the properties of the linear momentum^{1}.

To construct a quantitative theory to describe physical phenomena it is necessary to make use of some mathematical formalism in terms of which physical principles can be expressed so as to lead to quantitative relations between observables. Because the principle of uncertainty is entirely restrictive in its content, its role is primarily to restrict the mathematical formalism in terms of which the physical theory can be expressed. In principle, the nature of the formalism is arbitrary provided it allows for proper expression of the adopted principles and provided the axioms underlying it are consistent in a rigorous mathematical sense. The well known elements of the quantum formalism are reviewed here only to emphasize that they provide a remarkably simple basis for expression of the single notion that not all observables have exact values at the same time. In a classical theory, the formalism consists of relations between symbols representing the values of observables, and the symbols of the formalism therefore have the properties of mathematical variables. On the other hand, in a quantum theory that takes account of the principle of uncertainty, the formalism of the classical theory must be altered precisely because all observables do not have exact values in a given state of a physical system. Specifically, the formalism must be altered so as to prevent the assignment of precise values to observables that do not have precise values simultaneously. This necessary alteration of the formalism is accomplished most simply by expressing the relations between observables as operator relations in which observables are represented by operators (denoted O) that act on a symbol called a ket (denoted

Since the uncertainty principle disallows the possibility that all observables can have precise values simultaneously, it is necessary to distinguish between states in which certain observables have precise values and states in which all observables have only imprecise values. A state in which certain observables have precise values is referred to as an eigenstate of those observables, and the ket that represents it is referred to as an “eigenket”.The eigenket represents a listing of the observables and the values of those observables that specify the eigenstate.

In the development of the formalism it is found sufficient to define only one rule of operation for an operator acting on a ket, namely. the rule that determines the result of an operator corresponding to an observable acting on an eigenket of that observable. Given an observable a, and an associated operator

where

To connect the formalism to the results of physical measurements it is necessary to connect the elements of the formalism to real numbers. This is done by introducing quantities “conjugate” to kets, known as bras (and denoted

conjugate quantity written

number (denoted

The rule in Equation (3) is consistent with the basic rule of operation (2) whereby the expectation value of a in an eigenstate of a is reduced to its eigenvalue. To define all potential operations within the formalism it is necessary to include the rule of operation of an operator

operator. The rule is expressed by the equation

with real eigenvalues must be “self-adjoint” so as to satisfy the equation

The definition of the “value” of an observable in Equation (3) makes possible a quantitative definition of the uncertainty in the value of an observable. The uncertainty is defined in analogy with the conventional uncertainty in the result of measurements of an observable a, given by the “root-mean-square-deviation” of the meas-

ured values from the mean, written

rate measurements. Because the expectation value plays the role of the average value in the quantum formalism, we define the quantum mechanical uncertainty Da by the equality

where

It remains to express the principle of uncertainty quantitatively in the formalism. This is accomplished by defining a “superposition relation” for kets, which results in certain commutation relations for operators. Since the rules of the formalism given above determine only the operation of an operator on an eigenket of that operator, there exists arbitrariness in the formalism relating to the operation of an operator on a ket that is not an eigenket. In particular, it remains to prescribe the operation of an operator

Da_{k} and Db_{k}. The operation of _{k} about a mean value a_{k}, it follows that the operation of

A prescription for the operation of _{k}. The resultant definition is expressed by the equality

(and its counterpoint with a and b interchanged). Below, for simplicity, all kets are assumed to be normalized to unity and the eigenvalues a_{j} are taken to be discrete and to lie in the interval between

The extent to which superposition relations of the above form provide a proper expression of the principle of uncertainty must depend on the extent to which the equations resulting from the relations can be interpreted in a manner consistent with the principle. In particular, use of the relation (6) in the defining equation (3) must result in an expectation value of a consistent with the uncertainty in a. To compute this expectation value in the state represented by the ket

the fact that its product with

Here the normalization of all kets to unity guarantees the equality

Equations (7) and (8) show the expectation value of a in the state represented by the ket

The prescription for representation of the ket

Given the form and interpretation of the superposition relation (6), inclusion of terms in the superposition relation with zero coefficients (corresponding to zero probabilities) then allows summations over

A more quantitative expression of the uncertainty principle in the formalism is obtained through an equation for the value of the “commutator” of a pair of operators corresponding to conjugate observables. The equation can be developed from the definition of the uncertainty in an observable α in Equation (5) together with the expression for the expectation value in Equation (3).In particular, by multiplying together the uncertainties in observables α and b as expressed by Equation (5), and use of the definition of the expectation value in Equation (3), an expression for the product

where the bracket

consistent with the commutability of operators in the limit of zero

The relations (11) combined with Equation (2) make possible a re-expression of the operator Equation (10) as an ordinary equation connecting functions of α and b. The result is obtained by multiplication of Equation (10) on the left and right respectively by eigenbras and eigenkets of α to produce the relation

which Equations (2) and (11) (and choice of the “+” sign) reduce to the form

The extremely singular character of this last relation (for

then requires the quantity

The equivalence can be reworked into a more general relation (and its adjoint) by expansion of an arbitrary ket

The expansion makes it possible to rewrite Equation (14) as the more general relation

which (combined with the symmetry between α and b) serves to establish the well known prescriptions that define the result of an observable operator acting on a continuum eigenket or eigenbra of a conjugate observable

with the lack of complete symmetry between α and b a reflection of the particular choice of sign in the commutation relation of Equation (10).

The prescriptions (17), together with the rule (2), lead immediately to a differential equation for the bra-ket product

which has the solution

where C is an arbitrary complex constant, independent of both

The physical content of the formalism is increased by the addition of the principle of invariance. As an example of this, in the section below, the results obtained from this principle are derived in the simple case of invariance of the equations of the theory with respect to space translations. The results extracted from the invariance principle applied to the complete set of transformations in the Lorentz group are detailed in Reference [

The recognition of the existence of equivalent space-time frames relative to which the equations of physics should have the same form represents the essential notion underlying the “physical principle of invariance”. The content of the principle is then contained in the assertions that define certain distinct frames to be equivalent. Here, for simplicity, we focus only on the asserted physical equivalence of coordinate frames connected to one another by either a shift in the coordinate origin or a rotation of the coordinate axes. In either case, the space coordinates of a system with respect to the initial frame will be labeled x, y, z, while the coordinates of the same system with respect to the “transformed frame” will be labeled

In the ket picture, a transformation between an unprimed and a primed coordinate frame is asserted to leave the operators unaltered,

but is defined to result in a change in the bras and kets that describe the system by way of the prescriptions

with the quantity

where the change in the expectation value

but to result in a change in the operators according to the relation

In this case the expectation values of physical observables transform as

where the change in

Additional physical content is introduced into the definition of the transformation operators by the requirement of conservation of probability below. In the case of a linear transformation, this requirement restricts the transformation operator U to be unitary, as defined by the equation (see below),

in which case Equation (25) can be rewritten in the form

It is sufficient here to consider the case of continuous transformations that can be evolved from the identity operator I by a continuous sequence of transformations generated by infinitesimal variations in a parameter a. In this case, the transformation operator U can in general be represented as

where

We focus here on a transformation between the physically equivalent frames of a closed system resulting from an infinitesimal translation of the coordinate origin by an amount

where

Consider first the ket picture of the coordinate transformation in which the operator

where the kets

The result (32) combined with Equations (29), (30), and (31) is sufficient to determine the properties of the operator G, and in addition demonstrate that the observable associated with the related operator

The derivation proceeds on the basis of well known steps. By use of Equations (29), (30), and (31) the primed frame quantities

The functions of

By substitution of Equations (34) into Equation (33) and the neglect of terms proportional to the square of

which reduces to the result

The result is sufficient to establish that the observable

the form of Equation (36), in terms of the operator

But, since Equation (16) derives from the commutation relation (10) that represents an expression of the uncertainty relation

which in turn implies that the observable

That the operator

Consider now the operator picture of the infinitesimal translation in which an operator

By use of Equations (30) and (37) and the self-adjointness of the operator

The interest is in the connection between the operators corresponding to coordinate observables

lues of the coordinate operators are required to be related by Equation (29). It follows that the coordinate operators in the primed and unprimed frames,

from which it follows that the operator

operator

tion, which is valid when the operators in the equation act on any ket:

In particular, since the above connection between the operators

tors act on the eigenkets

tion between the operators must be valid when the operators act on any ket on which they can act. But as a consequence of Equation (41), the operators

and it follows from comparison of Equations (44) and (45) that the operators

Here the operator

by use of which Equation (46) can be reduced to an equality relating strictly to the degrees of freedom connected to the coordinate vector

It then follows from the fact that the derived commutation relation between the operators

is identical to the commutation relation between operators associated with conjugate observables, that the observables

The conclusion extracted from the operator picture of the coordinate translation is therefore the same as that deduced from the ket picture of the transformation (as it should be). In fact, because Equation (36)’ can be derived from Equation (46), the property of the operator

Similarly, by considering the connections between coordinate frames related by a rotation of the coordinate axes rather than by a translation of the origin, an analysis parallel to the above demonstrates that the observable associated with the (properly defined) generator of an infinitesimal rotation about the k-th coordinate axis,

with the operator associated with the observable

It can be demonstrated that the observables associated with the generators of infinitesimal translations and rotations have in fact all the familiar properties of linear and angular momentum observables [^{2} [

It is important that Equation (36)' in the form

where

The covariance of the equations under the complete set of Lorentz transformations requires that the two equations be combinable into a four-vector equation. This can be guaranteed only under the condition that the operators

But when an equation involves only simultaneous observables, the equation must be valid when the values of the observables are replaced by their corresponding operators acting on any ket

Multiplication of Equation (55) on the left by a coordinate eigenbra

which the notation

re-expresses as

An approximate equation for a free particle can be derived from this equation by solution of Equation (54) for the energy eigenvalue E in terms of a square root, and approximation of the square root in the case in which

The validity of Equations (58) and (59) is limited by the fact that the forms of the equations are altered by a phase change in the function

where

with

with

What is demonstrated here is that combination of the invariance principle with only a weak statement of the uncertainty principle that asserts only the existence of unspecified pairs of conjugate observables is sufficient to establish the existence of specific pairs of such observables, which in fact exhaust the complete set of known pairs of observables satisfying uncertainty relations of the form (1). It is shown in addition that the invariance principle, formulated within the quantum formalism, determines the forms of the basic “equations of motion” of the theory.

The author thanks Professor D. G. Ellis for useful comments relating to this paper.