The Role of Continuity in Quantum Mechanics

                 ___________________________________________

 

                               Daniel Stubbs

                    Department of Applied Mathematics

                      University of Western Ontario

                                London, ON

                                 N6A - 5B7

                        ds@pineapple.apmaths.uwo.ca

 

                         (received: July 9, 1996)

 



Abstract

________

 

	The prominent role and great importance of the concept of continuity, 

in a non-technical sense, for the development of classical mechanics, both at 

its birth in the work of Newton, and as it became the greatest physical theory 

of any age hitherto is well-known. The role that this same idea of 

``smoothness'' plays in quantum theory, in both the early formulations of 

Heisenberg and Schr\"odinger and in the later work dealing with relativistic 

fields of Dirac, Feynman and others, is considerably less well-understood. It 

is often asserted that one of the ways in which quantum mechanics represents a 

decisive break from classical mechanics, and in contrast to Einstein's theory 

of relativity which is formulated entirely in the language of differential, 

that is smooth, geometry, is by means of rejecting this axiom of continuity. 

This article will argue that continuity, though in a much more disguised 

fashion than in earlier theories, remains a vital force in quantum mechanics, 

and indeed quantum mechanics would be impossible without this concept of 

continuity.

 

1. Introduction

_______________

 

	The idea that nature is, in some sense, ``smooth'' is a very old one, 

having been expressed by Aristotle as *Natura saltum non facit* (Nature does 

not make a leap), and this idea continues to play a crucial role in scientific 

thinking right down to the present. In the quantitative physics of the modern 

age this Aristotlean dictum has been most commonly expressed in the demand 

that all quantities of physical importance be continuous, and in a more 

precise mathematical sense, differentiable. But with the advent of quantum 

mechanics in the early decades of this century many individuals, both 

physicists and philosophers, began to question the relevance of this axiom of 

continuity to contemporary physics. Quantum mechanics seemed to reject this 

traditional view of the smoothness and fluidity of physical processes, 

replacing it instead with discontinuous jumps or ``quantum leaps'' as they 

came to be called between physical states. The mathematics used to describe 

this strange new science of quantum mechanics was also different from that 

used in classical mechanics. Instead of the calculus of Newton and Leibniz, 

rectangular arrays of numbers called matrices were employed to calculate the 

frequency and height of these jumps between states. It became commonly felt 

that one of the many ways in which quantum mechanics represented a radical 

paradigm shift in physics was through this rejection of the classical belief 

in the continuity of physical processes. Einstein's theory of relativity 

remained classical not only because of its deterministic picture of the 

universe but also because of its use of differential geometry and conventional 

assumption that the universe can be thought of as a smooth four-dimensional 

manifold.

 

	In this paper it will be argued that this portrait of quantum 

mechanics and the role of continuity in quantum mechanics suffers from several 

shortcomings. The role of continuity is much more subtle and disguised than in 

classical mechanics and relativity theory but quantum mechanics would be as 

crippled as either of these two other theories were the assumption of Nature's 

smoothness to be called into question. Rather than the directly observable 

physical variables themselves being smooth, other non-physical quantities such 

as the wavefunction of Schr\"odinger's equation are assumed to be 

differentiable and even the seemingly radical new mathematics of Heisenberg's 

matrix mechanics can be seen to be embedded within the traditional 

``mathematics of continuity'', a mathematics that had its start with the work 

of Newton, Leibniz and others in the 17th and 18th centuries. Newton's 

invention of calculus went hand in hand with his invention of dynamics and 

universal gravitation and even his choice of terms for his new mathematical 

quantities, *fluxions* and *fluents*, are charged with the idea of 

quantities flowing smoothly, like water, both of them derived from the Latin 

verb *fluere*, ``to flow'' as Newton well knew. It was Leibniz who first 

gave the fullest expression to the scientific potential of this axiom of 

continuity and argued for its explicit use throughout scientific 

investigations:

 

		The description of nature in continuous and differentiable

	functions now becomes the primary postulate; the continuity principle

	becomes the basic presupposition of all exact knowledge of nature.

	For Leibniz it is not an empirical principle deducible from, and

	capable of being reduced to, individual observations. Rather he

	introduces it as a general principle, a principle of universal

	order (*principe de l'ordre g\'en\'eral*). [...] This principle

	is not simply a law of nature; rather it is a universal law showing

	us how we are to find and grasp natural laws. [...] Only on the basis

	of this postulate can the newly achieved instrument of knowledge, the

	analysis of the infinite, be made to serve the investigation of

	nature and to be truly fruitful there.

		[Cassirer pp. 158-159]

 

However, the importance of continuity as a tool of mathematical physics was 

not fully seized upon until the mid and late 18th century. It was then that 

mathematicians like D'Alembert and Lagrange began to make the action principle 

of the variational calculus, and the systems of differential equations which 

it made the official language of theoretical physics, the central focus of 

scientific thinking.

 

	By the time Kant wrote his *Critique of Pure Reason*, Leibniz' law 

of continuity had been elevated to the status of a fundamental characteristic 

of nature. Space and time, and in consequence variables of motion dependent 

upon space and time such as velocity and acceleration, were believed to be 

definitively and absolutely continuous:

 

		The property of magnitudes by which no part of them is the

	smallest possible, that is, by which no part is simple, is called

	their continuity. Space and time are *quanta continua*, because no

	part of them can be given save as enclosed between limits (points

	or instants), and therefore only in such fashion that this part is

	itself again a space or a time. Space therefore consists solely of

	spaces, time solely of times. Points and instants are only limits,

	that is, mere positions which limit space and time. But positions

	always presuppose the intuitions which they limit or are intended

	to limit; and out of mere positions, viewed as constituents capable

	of being given prior to space or time, neither space nor time can be

	constructed. Such magnitudes may also be called *flowing*, since

	the synthesis of productive imagination involved in their production

	is a progression in time, and the continuity of time is ordinarily

	designated by the term flowing or flowing away.

		[Kant, p. 204]

 

In this passage, we see how for Kant the axiom of the continuity of space and 

time had come to be identified with the infinitesimal calculus of Newton and 

Leibniz through the crucial concept of a *limit*. A point of the 

continuum, for instance `0', is comprehended as the accumulation point of a 

sequence, e.g. `0' is the accumulation or limit point of the sequence `1/n' as 

`n -> infinity'. This, of course, is nothing other than the canonical 

construction of the real numbers as the completion of the set of Cauchy 

sequences of rational numbers.

 

	Even the realization later on in the 19th century that continuity and 

differentiability are not synonymous and that for a function `f(x)' to have 

one derivative `f'(x)' does not mean that it has all derivatives 



	 (n)

	f  (x),  n = 2,3,...



or can be expanded as a convergent power series 

 

      (infinity)

	__           n

	\    a  (x-c)   ,

	/_    n

      (n=0)



did not alter the fundamental role played by 

calculus in physics. Physicists simply assumed as a matter of course that all 

functions of physical interest were analytic functions, i.e. expandable as a 

convergent power series in some open neighbourhood of the region of physical 

interest for the independent variable(s), of all variables and/or parameters 

deemed to be relevant to the necessary calculations. Nor did the gradual 

transition from a mechanical concept of Nature to a field theoretic conception 

of the universe affect the implicit assumption of analycity; partial rather 

than ordinary differential equations were used in the modelling process and 

while this makes the actual construction of explicit solutions far more 

delicate and difficult than it had been previously it had no effect on the 

issue of discreteness versus smoothness.

 

	The advent of Einstein's theory of relativity in 1905, while 

profoundly altering many of science's concepts of space, time and, in the 

general theory, gravitation also left one crucial Newtonian concept unchanged, 

namely the continuum character of space and time. The manifold of spacetime 

was assumed to be smooth, indeed the elements of the metric tensor are 

conventionally taken to possess derivatives of all orders, and locally 

Euclidean, which allowed one to think of the curvature in a perturbative, 

Taylor series-like manner. The field equations themselves are local partial 

differential equations of the second-order in the metric tensor, and the 

geodesic equation describing the motion of a test particle is derived from an 

integral variational principle. In this way, as in the deterministic nature of 

the differential equations, relativity theory was seen as a part of classical 

physics. But only a few years after the publication of Einstein's papers on 

relativity in 1905 and 1915, a new mechanics of the atom emerged that was to 

throw into question these ideas of determinism and, it seemed, the continuity 

of natural phenomena.

 

2. Discreteness and the Early Quantum Theory

____________________________________________

 

	From the first step on the path to the eventual finished formulation 

of quantum mechanics it seemed that discreteness was destined to play a very 

large role in this new theory; indeed, it was precisely in advancing the 

hypothesis of discreteness that quantum mechanics broke away from the 

classical domain. The radiation emission pattern of a blackbody could only be 

realized theoretically by assuming that the radiative energy was emitted 

discretely, in quanta, whose energy was given as a function of frequency by 

Planck's famous law ` E = h nu' , and not continuously as had been thought. 

Einstein's paper on Brownian motion in 1905 provided direct empirical evidence 

for the existence of atoms, suggesting that matter at a fundamental level was 

particulate and not smooth, and the only way that Bohr could prevent the 

hydrogen atom from collapsing in upon itself and radiating energy continuously 

was by assuming that the orbiting electron did not obey the (differential) 

equations of Maxwell and Newton but instead jumped from one distinct energy 

level to another, and was unable ever to jump below the ground state of 

-13.6 eV.

 

	But it was in the work of Heisenberg that discreteness seemed to loom 

the largest, leading even to what was felt to be a new mathematics of matrices 

and linear algebra radically different from the calculus of Newton and 

Leibniz. Heisenberg's work focussed exclusively on physical observables, such 

as the energy, and represented an `n' state system as an `n x n' matrix, 

of which the most important example remained the hydrogen atom, a system with 

a denumerably infinite number of states, and the most important task that of 

reproducing its spectrum, particularly in the case of the line splitting that 

occurs when a hydrogen atom is placed in a constant magnetic field, the 

so-called Zeeman effect. In this new matrix mechanics, there were no smooth 

functions, no derivatives, no differential equations, the science of quantum 

mechanics seemed to consist of the purely algebraic tasks of multiplying two 

matrices together (introducing the novel problem of non-commutativity into 

physics), finding the eigenvalues of matrices and diagonalizing them over 

various fields, usually just the field of complex numbers. These eigenvalues 

that occurred in the diagonal should then correspond to numbers actually 

observed in experiments.

 

	In retrospect though, we can see that Heisenberg's matrix mechanics 

was not, perhaps, as radical as was originally thought. All of the 

calculations were performed over the fields of real or complex numbers, that 

is, fields whose cardinality was that of the continuum. Neither was linear 

algebra as new or as radical to mathematicians as it was to physicists; the 

idea of matrices had begun with the work of Sylvester and Cayley in the 1860s 

and 1870s, over fifty years prior to Heisenberg's work, see [Boyer, pp. 

627-30]. As the work of Hilbert and others was to demonstrate, 

infinite matrices could be used to represent linear differential operators 

acting on an infinite dimensional vector space of functions. Lastly and 

perhaps most importantly was the inherent linearity of matrices, a fact that 

tended to diminish whatever discreteness may have existed in Heisenberg's 

mechanics. A linear operator is one whose output is, in some abstract metric, 

linearly proportional to its input; if `x' is a vector in R^n and `A' is 

some linear mapping from R^n to R^n then we know that ~x = A(x) can be written 

as |~x| = A|x| where `A' is simply a constant number. The modern definition of 

continuity in terms of (delta) and (epsilon) reads, a function f(x) is 

continuous as `x' approaches `a' if for every (epsilon) > 0 there exists a 

(delta) > 0 such that |x-a| < (epsilon) implies that |f(x)-f(a)| < (delta). 

Clearly, the function from R^n to  R defined by F(|x|) = A|x| satisfies this 

definition of continuity so long as the metric |.| is well-behaved (such 

as the Euclidean metric which is smooth except at 0). Because of the use of 

linear operators throughout, Heisenberg's matrix mechanics is in fact far less 

discontinuous than might at first glance be thought. Linearity is a property 

that inherently imposes a degree of smoothness and proportionality between the 

variables, and thus excludes the most severe sorts of discontinuity. 

Denumerable sequences of real numbers were nothing new for physicists either, 

the work of Strum and Liouville in the 1840s had made it clear that any 

well-posed boundary value problem for a linear homogeneous second-order 

ordinary differential equation would lead to a denumerable set of eigenvalues 

(lambda_n) for which the boundary value problem would possess non-trivial 

solutions. These mathematical results had been used extensively in solutions 

of linear partial differential equations in everything from heat diffusion to 

the vibration of membranes throughout the late 19th century.

 

	Finally, Heisenberg's matrix mechanics was only being used on bound 

systems, essentially atoms, and only very simple ones at that, such as 

hydrogen, helium and so forth. There seemed to be no obvious manner in which 

to generalize the method of matrices to scattering phenomena, one of the chief 

means by which atomic structure had been investigated by Rutherford and 

others, or to much more complex atoms like those of iron, gold or uranium. And 

despite the success of Heisenberg's approach in explaining the spectral 

structure of hydrogen, many physicists remained uncomfortable with the use of 

such strange tools as matrices and the seeming lack of any intuitive 

visualization for the physics of matrix mechanics. In the same year as 

Heisenberg's matrix mechanics was published, so were the papers of an Austrian 

theoretical physicist whose work would be the most responsible for the 

acceptance of quantum mechanics by the broad majority of physicists, despite 

the fact that he was destined to regret his involvement in the quantum 

revolution.

 

3. The Revenge of Continuity: 

_____________________________

Schr\"odinger's Wave Mechanics and Quantum Field Theory

_______________________________________________________

 

	Schr\"odinger's approach to quantum mechanics was through the 

thoroughly classical concept of waves, familiar to generations of physicists 

from experiments with light, water and sound and well understood 

mathematically through the wave equation,

 

                           1

                    u   - ---   u     = 0

                     xx   c**2   tt 



first extensively investigated by D'Alembert in the 18th century. 

Schr\"odinger's 

work grew out of the optical research of Hamilton and others in the late 19th 

century, and used their thinking about how waves of light behaved in various 

media to begin his own work about how the ``matter waves'' of De Broglie might 

be represented by a partial differential equation similar to the wave equation 

of D'Alembert. Schr\"odinger's eventual formulation required the use of a 

complex-valued function that had no immediate physical interpretation and was 

rather vaguely described as the ``wavefunction'', but what mattered for most 

physicists was that the whole affair could quickly be reduced to a linear 

second-order partial differential equation, 



                                       2

                      d Psi        hbar 

              i hbar  _____  =  - _____ (del) Psi  + V  phi

                       d t         2 m



If the potential function depended only on the spatial variables, as was 

often the case (or at least served as a useful approximation), then the 

simple separation of variables Psi(t,x) = psi(x}) exp(i hbar E t) 

removed the mysterious factor of `i' and converted quantum mechanics to an 

exercise in the construction of solutions to the Helmholtz equation subject to 

appropriate boundary and initial data: 



                        2 

                    hbar       2

                -  ______ (del)  psi + (E - V(x)) psi = 0   .

                     2 m



While the wavefunction Psi remained as physically 

inexplicable and unvisualizable as Heisenberg's matrices, it was quickly 

realized that its absolute value |Psi(t,x)|^2 functioned much like a 

probability density and thus in one dimension for instance the expression 

|Psi(t,x)|^2  dx could be interepreted as the probability of the 

particle represented by Psi being located between `x' and `x+dx'. The 

approach taken by Schr\"odinger to quantum mechanics was what finally took 

quantum mechanics from being an esoteric and bizarre branch of physics, a 

plaything of theoreticians that was poorly understood by the broad majority of 

physicists, to being what it is today, an indispensable tool of working 

scientists in almost every branch of physics, chemistry and astronomy. By 

showing physicists the manner in which quantum mechanics could be seen as just 

a toolkit of differential equations, to be solved as physicists had solved 

them for two hundred and fifty years. Schr\"odinger's approach lent itself 

naturally to perturbation methods for dealing with complex atoms like iron and 

uranium, one simply used the standard perturbation methods developed for the 

complex differential equations of fluids and celestial mechanics, and the 

difference between bound and unbound systems, or atoms versus scattering 

phenomena, could be handled by simply assuming that the energy parameter $E$ 

was greater than than 0 (if V(x) ->  0 as |x| -> infinity ). While 

the matrix approach of Heisenberg still makes sense for very simple systems 

(e.g. a two-state system of spin up and spin down), and the matrix-oriented 

notation of Dirac is used quite regularly in advanced texts on quantum 

mechanics, it is the smooth calculus-oriented picture of Schr\"odinger that 

constitutes the way in which most physicists conceive of quantum mechanics and 

work with it on a daily basis.

 

	With the advent of the quantum theory of systems possessing an 

uncountably infinite number of degrees of freedom beginning in the 1930s, most 

of the last notions of quantum mechanics as a science that stood apart from 

the concept of continuity, a science that depended on a radically new and 

different mathematics than that of the traditional classical analysis of 

integral variational principles, differential equations and so forth, were 

quietly forgotten. The new theory of relativistic quantum fields that was 

pioneered by Dirac, Feynman, Pauli and Schwinger, to name a few, was cast 

entirely in a language familiar to generations of physicists, indeed the 

variational approach had its beginning with Lagrange's *Analytic 

Dynamics*. Due to the fact that the quantum theory of interacting fields 

results in nonlinear equations, perturbative expansions in a coupling constant 

became common and thus it was implicitly assumed that not only did the 

relevant physical quantities (in this case, scattering amplitudes) depend 

smoothly on the space and time variables but also on the coupling constant. 

When it was found that these series were in fact divergent, it was then 

assumed that they were asymptotic in nature and certainly the incredible 

accuracy of the predictions of the new field theory, such as of the magnetic 

moment of the electron, lent credence to this view. But the whole mathematical 

apparatus of modern quantum field theory would seem to represent a triumph for 

the spirit of Newton and Leibniz, for the language of infinitesimals dominates 

throughout and it would seem that Leibniz' *principe de l'ordre 

g\'en\'eral* remains just that, a universal principle of the physical world 

without which we would be crippled in our attempts to quantitatively 

understand this universe. Not only the language but also the way of thought 

that is embodied in calculus, be it at the elementary level practised by 

Newton or at the level of multiple scale analysis for nonlinear partial 

differential equations. This way of thought is inherently reductionist, the 

use of differential equations automatically implies the existence of a causal 

principle of some kind --- every differential equation requires the 

specification of various initial/boundary data in order to uniquely determine 

a solution --- and, particularly in the most modern evolution of quantum 

mechanics, the great prominence of perturbative series expansions encodes the 

idea that physical processes can be built up step by step in the fashion of a 

series. Nature is just a smooth and gentle deformation of nothingness and the 

truth can be grasped by smoothly running our minds over a series of  

*naturae simplices*, in the words of Descartes.

 

	The considerable mathematical difficulties that presently beset 

certain aspects of both quantum field theory and Einstein's general theory of 

relativity, most notably the issue of singularities and ``infinities'', 

whether they be swept under the rug by means of renormalization or (it is 

hoped) hidden behind the curtain of an event horizon, cannot help but leave 

one curious as to whether a truly discontinuous physical theory, a theory 

based on the assertion that *Natura saltum semper facit* and that would of 

necessity involve a genuinely non-analytic mathematics, could have avoided at 

least some of the mathematical quagmire that the Standard Model is presently 

stuck in. No two physical theories in the history of science have been more 

accurate than quantum electrodynamics and general relativity, and yet no two 

theories have been less sound from the viewpoint of mathematical rigour than 

these two. Certainly many phenomena treated by contemporary physics does seem 

to behave in a highly discontinuous manner, most particularly in high-energy 

physics. The constant creation and annihilation of pairs of particles and 

their spontaneous disintegration into showers of other particles are both 

well-known examples of the way in which many modern physical systems seem to 

delight in jumping from state to state, even from being into non-being, in 

times of far less than a second. Despite this discontinuity in the empirical 

world, the mathematics used in modern quantum mechanics is of a kind with that 

of Newton and Leibniz and indeed represents perhaps the greatest triumph of 

all for classical physics, that while the concepts it employed might be 

discarded, the language in which those concepts were expressed remains the 

``official language'' of physical science over three hundred years after the 

publication of the *Philosophiae naturalis principia mathematica*.

 

	In conclusion then, we have seen that the role of continuity in 

quantum mechanics has been an evolving one, from an initial belief that 

Heisenberg's matrix mechanics represented a radical break from the Newtonian 

tradition of continuity to the invention of the wave mechanical approach to 

quantum mechanics and continuing up until recent developments in quantum field 

theory. With each advance the perceived break that quantum mechanics was from 

classical mechanics, at least in terms of the continuous versus discontinuous 

debate, became less and less; quantum mechanics, viewed in terms of the 

wavefunction Psi(x,t) and its differential equation, began to seem 

far less mysterious and inscrutable, on a mathematical level (on an 

epistemological level it has lost none of its mysteriousness, even after 

seventy-five years of debate and controversy) and became a tool that was 

accessible to the majority of physicists, trained in the classical tradition 

and familiar with only a very limited stock of mathematical tools and 

techniques (that only rarely included the linear algebra needed to appreciate 

Heisenberg's work). And it was only by securing the consensus of this broad 

majority of physicists that quantum mechanics was able to elevate itself from 

a theoretical curiosity into the centerpiece of 20th century physics. Quantum 

field theory, the contemporary descendant of the elementary non-relativistic 

quantum mechanics of Heisenberg and Schr\"odinger, has become even more 

closely associated with analysis, most notably in the use of singular 

perturbation theory to calculate the Green's functions of systems of nonlinear 

partial differential equations arising out of various Lagrangian models of 

interacting fields. The concept of continuity has come full circle in quantum 

mechanics, its apparent absence in 1925 being hailed as one proof of the 

radically new nature of the quantum theory, but now is rests at the very heart 

of the variational calculus and perturbation theory that is the language of 

quantum field theory.

 

References

__________

 

[1] Cassirer, E., "Determinism and Indeterminism in Modern Physics", 

	Trans. O. T. Benfey. New Haven: Yale University Press, 1956

 

[2] Kant, I. "Critique of Pure Reason", Trans. N. K. Smith.,

	London: Macmillan, 1929

 

[3] Boyer, C. "A History of Mathematics", 

	Princeton: Princeton University Press, 1968