- Email: [email protected]
The Dirac equation for accountants
Chaos, Solitons and Fractals 30 (2006) 407–411 www.elsevier.com/locate/chaos
The Dirac equation for accountants G.N. Ord
*,1
Perimeter Institute for Theoretical Physics, Waterloo, Ont., Canada Accepted 9 January 2006
Abstract In the context of relativistic quantum mechanics, derivations of the Dirac equation usually take the form of plausibility arguments based on experience with the Schro¨dinger equation. The primary reason for this is that we do not know what wavefunctions physically represent, so derivations have to rely on formal arguments. There is however a context in which the Dirac equation in one dimension is directly related to a classical generating function. In that context, the derivation of the Dirac equation is an exercise in counting. We provide this derivation here and discuss its relationship to quantum mechanics. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The Dirac equation unites the two fundamental theories of special relativity and quantum mechanics. Given the complexity and counter-intuitive nature of the physical phenomena being described, the compact representation by a single partial differential equation suggests that the language involved has considerable power. However the power that enables the description of complex phenomena can also obscure simplicity. There is a case to be made that if we start with simple language, we may be better able to interpret conventional formulations. In this article, we shall trade our mathematical toolkit for that of an accountant. We shall allow only addition, subtraction, multiplication and division aspoperations suitable for precise book-keeping. We shall abandon the common ffiffiffiffiffiffiffi number systems of complex numbers ( 1 is difficult to interpret directly in terms of counting) and even real numbers (impossible to add or record) in favour of integers, or where necessary, rational numbers. This means that we lose the differential calculus; a pretty drastic loss considering its central presence in the language of physics. Still, the idea is that if we are able, in the end, to understand a discrete approximation to the Dirac equation through simply counting recognizable objects, we shall have gained a new understanding of this enigmatic partial differential equation. The Dirac equation is usually produced by arguments that begin by requiring a partial differential equation of the Schro¨dinger form ih
*
1
ow ¼ H w. ot
Tel.: +1 4169795000x6967; fax: +1 4169795336. E-mail address: [email protected] On leave from the Department of Mathematics, Ryerson University, Toronto, Ont., Canada.
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.052
ð1Þ
408
G.N. Ord / Chaos, Solitons and Fractals 30 (2006) 407–411
This is followed by the relativistic requirement that E2 ¼ m2 þ p2 ;
ð2Þ
where m is the rest mass of the electron and p is the momentum. Combining these requirements lead Dirac to ih
ow ¼ ða p þ bmÞw; ot
ð3Þ
with suitable anticommutation relations for the matrices a and b. Much as the original argument was brilliant and insightful at the time, there was no sense in which the resulting equation described a wavefunction w that had a prior physical meaning. As in the non-relativistic case, w here is an abstract tool in a probability calculus. The objective of the calculus is to obtain expectation values of ‘observables’ using w. Whether w has any physical analog in Nature is unknown to this day. Now solutions of (3) involve wave propagation, spin, the uncertainty principle, special relativity, and a host of features we commonly associate with relativistic quantum theory. It is natural to assume then that the equation itself is intimately related to some pretty complicated physics. How then are we to extract such an equation from a simple accounting problem? The trick is to look beneath the elegant mathematical structure of the differential equation for a simple statistical mechanical picture.2 With such a picture in place we shall see that we can indeed understand the equation in terms of simple accountancy tools. The added bonus will be that as accountants, we shall always know what we are counting; we shall know what w represents.
2. The Entwined path model We consider a special ‘‘entwined’’ walk on a square lattice in the (x, y)-plane [8–10]. The lattice spacing will be some small positive rational number and the walker will move at each step in one of the four directions (±1, ±1). The stochastic element of the walk is governed by the binomial distribution. That is, at step (n + 1), the walker will step in the same direction as in step n with probability (1 m), with m a fixed positive constant and m 1. The alternative at each step, besides continuing in the same direction, is a change of state. A change of state occurs in an alternating pattern. The walk is started from the origin and the first step is to (1, 1). The stochastic process is consulted and the walk continues to (2, 2) unless a state change is indicated. The first state change is a change in the direction of motion along the x-axis. So if this happens at (1, 1), the subsequent step is to (0, 2). The next state change will not change the direction of the walk, but the walker will drop a marker for its return path before stepping to the next site. Thus all the odd numbered state changes will be direction changes in the x coordinate, all the even numbered state changes will preserve direction but drop markers. This process will continue until the walk steps past some pre-assigned y-coordinate, yM say, at which point, at the next marker drop, the walker will return to the origin along the path defined by the set of markers. A single such entwined path (EP) is illustrated in Fig. 1. Notice that the EP defines a chain of oriented rectangles in which the orientation reverses at each crossing point of the EP. If we use the ‘right hand rule’ for path traversal, the first rectangle has positive orientation out of the page, the second negative, into the page, and so on in an alternating fashion. As the walk is lengthened by repeating the process we can imagine that a cone with apex at the origin will become covered by an ensemble of these oriented rectangles. If we sit at a particular off-lattice point inside this cone we could expect to be regularly encircled by these oriented rectangles and we might expect that we should eventually see an equilibrium distribution of orientation, varying from point to point throughout the cone. How could we measure this orientation? Consider again Fig. 1. Each oriented rectangle has a left and right boundary that contributes to the oriented area. So as to avoid counting the contributions of an area twice let us restrict our attention to the right boundary of each rectangle. The collection of right boundaries itself forms a path that we call the ‘enumerative path’. Note that the enumerative path has a very regular structure and we can use it to count oriented rectangles. The blue segments of the path belong to positively oriented rectangles and add a + 1 to the oriented area count. The red portions of the enumerative path belong to negatively oriented rectangles and contribute a weight of 1 to the count. Now our task to count oriented rectangles is reduced to counting the contributions of enumerative paths on the lattice. We can do this using simple combinatorics, or we can assume an equilibrium is reached and use the structure of the walks to deduce what the equilibrium pattern must be. The latter route is the simplest.
2
This idea was initiated by Feynman in what became known as his ‘Chessboard model’ [1–7].
G.N. Ord / Chaos, Solitons and Fractals 30 (2006) 407–411
409
+
–
+
–
+
Fig. 1. On the left is a single ‘entwined path’ in the (x, y) plane. The colour indicates the direction of traversal, blue for traversal in the +y direction, red for traversal in the y direction. The origin is at the bottom of the path and the lattice spacing is small on the scale of the figure. Notice that the crossing of the forward (blue) and backward (red) paths forms a chain of oriented areas. The orientation switches from one rectangular area to the next. This is illustrated by the recolouring of the path in the centre figure. There blue represents positive orientation and red represents negative orientation. To count oriented rectangles we can simply use the ‘enumerative’ path, illustrated on the right hand side of the figure. Here the colouring illustrates the contribution to orientation, +1 for blue, 1 for red. (For interpretation of the references in colour in this figure caption, the reader is referred to the web version of this article.)
Let us label the lattice sites by x = m, and y = n where m and n are integers. We consider a two-component density u±(x, y) where u+ counts the number of (1, 1) and (1, 1) directed links and u counts the number of (1, 1) and (1, 1) directed links by orientation. We need a two-component density here because our enumerative paths continually shuffle orientation counts between the two directions. Now any link at (x, y + ) in the ±(1, 1) direction either follows a link of the same direction and colour at (x , y) or follows a link of the opposite direction and colour at (x + , y). The former occurs with probability 1 m, the latter with probability m. Thus if an equilibrium density is reached it must satisfy uþ ðx; y þ Þ ¼ ð1 mÞuþ ðx ; yÞ mu ðx þ ; yÞ.
ð4Þ
Notice here the subtraction involved in the second term. This is because whenever our enumerative path ‘turns right’ it switches orientation, thus changing the sign of its contribution. Since this happens for all paths, it must happen for the equilibrium distribution. We can similarly deduce that the u density must obey the difference equation u ðx; y þ Þ ¼ ð1 mÞu ðx þ ; yÞ þ muþ ðx ; yÞ.
ð5Þ
The positive sign for the second term reflects the fact that the change of direction for a ‘left turn’ on an enumerative path does not change orientation. Eqs. (4) and (5) are difference equations that, with suitable initial conditions for the u± yield rational number solutions inside the cone above the x-axis. The equations themselves are just conservation equations that express the fact that the enumerative paths are continuous and have the alternating orientation of Fig. 1. The solutions u± spread out into ever larger regions along the x-axis as y increases, so u includes an exponential decay. We may choose to look through this background decay to see the emerging pattern by following the evolution of: w ðx; nÞ ¼ u ðx; nÞð1 mÞn ;
ð6Þ
which satisfies the equations wþ ðx; y þ Þ ¼ wþ ðx ; yÞ mw ðx þ ; yÞ; w ðx; y þ Þ ¼ w ðx þ ; yÞ þ mwþ ðx ; yÞ to lowest order in .
ð7Þ
410
G.N. Ord / Chaos, Solitons and Fractals 30 (2006) 407–411
Let us now pause and note that Eq. (7) is a perfectly respectable equation for an accountant to produce. There are no abstractions or mathematical tools involved that take us out of the domain of good accountancy. We know exactly what is being counted . . . oriented rectangles produced by a long entwined path. We know the number system involved . . . the counting actually just used integers but the use of probability and normalization placed us in the domain of rational numbers. We have assumed that an equilibrium pattern will emerge from our initial conditions but that is something that is easily checked by performing a numerical experiment. Indeed, the calculation up to this point involves only a basic counting of recognizable objects on a planar lattice. Now physicists are generally impatient with difference equations and restrictions to rational numbers. So now let us be physicists once more and do something an accountant is unlikely to do, namely idealize the situation by taking a continuum limit. If we subtract w±(x, y) from both sides of (7), divide by and take the limit as ! 0+ we find that oy wþ ðx; yÞ ¼ ox wþ ðx; yÞ mw ðx; yÞ; oy w ðx; yÞ ¼ ox w ðx; yÞ þ mwþ ðx; yÞ 1 0 0 1 wþ ; rz ¼ and rq ¼ ¼ iry we have or writing w ¼ w 0 1 1 0 oy w ¼ rz ox w imry w.
ð8Þ
ð9Þ
This may be recognized as a form of the Dirac equation where c = h = 1 and y = t. Note that if we iterate this equation to get a second order form we have o2y w ¼ o2x w m2 w;
ð10Þ
which is the Klein-Gordon equation. Notice that to get (9) and (10) we have not invoked: (A) (B) (C) (D) (E)
the uncertainty principle, quantization or Schro¨dinger’s equation, complex numbers, special relativity, any ‘interpretation’ regarding quantum mechanics and the nature of reality.
We have simply taken the output of an ‘accounting argument’ and written it in a language familiar in the context of relativistic quantum mechanics. Now (9) is just a continuum limit of (7) written in a familiar form. (Notice that there has been no analytic continuation forced on the system, w is real and the i in (9) is present only because ry is imaginary.) The point here is that we can regard (9) either as a fundamental equation about the ‘wavefunction’ of an electron, without knowing exactly what a wavefunction represents in the physical world, or we can take (9) as the continuum limit of an equation describing an equilibrium distribution of a simple stochastic process. The continuum language that we use does not tell us whether we are describing a ‘Dirac wavefunction in one dimension’ or a ‘spacetime that maintains an accountancy ledger for the EP stochastic process’. Since Dirac wavefunctions are rather mysterious objects, it is worthwhile going through the above list to see how each property is manifest in the stochastic model. Here is a brief summary. (A) The uncertainty principle. This is built into the accountancy model by the fractal geometry of the EP’s themselves. If we look at a single enumerative path on scales much greater than 1/m we will see that it scales as a random walk in x, giving a fractal dimension of 2. This gives the correct scaling for the uncertainty principle. (B/C) Oscillatory solutions. The form of the accountancy problem solutions arises from the fact that we are counting oriented areas. Had we not recognized orientation in the counting, the minus sign on the right in (4) would have been a plus sign and the matrix iry in (9) would be replaced by rx. With that replacement, both (9) and (10) would become forms of the telegraph equations, which describe diffusive processes. Thus the ‘quantum’ aspect of oscillatory solutions and the implication of complex numbers all come from the counting of oriented areas, signaled by the presence of the matrix rq. Although this matrix is real, it has eigenvalues ±i which account for the oscillatory nature of the solutions. Dx (D) Special relativity. The paths in the stochastic model move on the diagonal so that at each step Dy ¼ 1. This translates to Dx ¼ 1 in the Dirac equation. Thus the constancy of the speed of light c = 1 is respected in the Dt model. The interpretation of a ‘Lorentz transformation’ is more subtle, involving both the geometry of the paths and the algebraic properties induced by orientation.
G.N. Ord / Chaos, Solitons and Fractals 30 (2006) 407–411
411
(E) Interpretations. The accountancy problem received no input from any particular view of a ‘quantum mechanical reality’. When we set up the problem, we did so assuming a classical distinction between object (the EP) and the observer (the accountant). There was no sense in which these were mixed. The random walker left a trail on the (x, y)-plane and the accountant simply counted links in the path by orientation. We did not have to discuss a ‘state of knowledge’ of the accountant, nor was the walker obliged to interact with the accountant. The wavefunction solutions of the difference equation had no apparent dependence on, or connection to the accountant’s consciousness, or for that matter to multiple universes, pilot waves or ensembles of similar systems. In the accounting problem the wavefunction was a manifestation of what J.S. Bell could have called a ‘beable’ . . . the EP. Since the wavefunction in the accounting calculation has an unambiguous interpretation, this provides an illustration that the problems of interpretation in quantum mechanics stem not from the unitary evolution of the wavefunction, but from the measurement postulates. Given that it is generally understood that measurement, not propagation is the prime difficulty for interpretations of quantum mechanics, is there something to be gained from this demonstration? I think the answer is yes! Notice that to see that there is an interpretation of Eq. (9) that can be explained in simple terms to an accountant, we have to look at the dynamical process and show how the act of counting oriented areas leads, in equilibrium, to a difference equation that may in turn be approximated by a differential equation. We had to see how counting, equilibrium and a continuum limit all conspired to give us a compact language to describe the final result. Had we started with solutions of the Dirac equation and been asked to interpret them for an accountant, we would have been hard pressed to do so. The ‘natural’ interpretation of solutions of the Dirac equation is in terms of waves. These are not features of entwined paths themselves but objects constructed by them. By comparison, in the interpretation of quantum mechanics we usually take wavefunction behaviour as a starting point for discussions of experiments on real systems. The accountancy problem shows that this may limit our view of what is happening in nature. Our mathematical language that works well for the calculation of statistical averages may be less than effective at revealing processes underlying those averages.
Acknowledgement The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Feynman RP, Hibbs AR. Quantum Mechanics and Path Integrals. New York: McGraw-Hill; 1965. Gersch HA. Int J Theor Phys 1981;20:491. Jacobson T, Schulman L. J Phys A 1984;17:375. Ord GN. J Stat Phys 1992;66:647. Ord GN. Chaos, Solitons & Fractals 1996;7:821. Gaveau B, Jacobson T, Kac M, Schulman LS. Phys Rev Lett 1984;53:419. Kauffman LH, Noyes HP. Phys Lett A 1996;218:139. Ord GN, Gualtieri JA. Phys Rev Lett 2002;89. Ord GN, Mann RB. Phys Rev A 2003;67. Ord GN, Mann RB. Ann Phys 2003;308.
The Dirac equation for accountants G.N. Ord
*,1
Perimeter Institute for Theoretical Physics, Waterloo, Ont., Canada Accepted 9 January 2006
Abstract In the context of relativistic quantum mechanics, derivations of the Dirac equation usually take the form of plausibility arguments based on experience with the Schro¨dinger equation. The primary reason for this is that we do not know what wavefunctions physically represent, so derivations have to rely on formal arguments. There is however a context in which the Dirac equation in one dimension is directly related to a classical generating function. In that context, the derivation of the Dirac equation is an exercise in counting. We provide this derivation here and discuss its relationship to quantum mechanics. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction The Dirac equation unites the two fundamental theories of special relativity and quantum mechanics. Given the complexity and counter-intuitive nature of the physical phenomena being described, the compact representation by a single partial differential equation suggests that the language involved has considerable power. However the power that enables the description of complex phenomena can also obscure simplicity. There is a case to be made that if we start with simple language, we may be better able to interpret conventional formulations. In this article, we shall trade our mathematical toolkit for that of an accountant. We shall allow only addition, subtraction, multiplication and division aspoperations suitable for precise book-keeping. We shall abandon the common ffiffiffiffiffiffiffi number systems of complex numbers ( 1 is difficult to interpret directly in terms of counting) and even real numbers (impossible to add or record) in favour of integers, or where necessary, rational numbers. This means that we lose the differential calculus; a pretty drastic loss considering its central presence in the language of physics. Still, the idea is that if we are able, in the end, to understand a discrete approximation to the Dirac equation through simply counting recognizable objects, we shall have gained a new understanding of this enigmatic partial differential equation. The Dirac equation is usually produced by arguments that begin by requiring a partial differential equation of the Schro¨dinger form ih
*
1
ow ¼ H w. ot
Tel.: +1 4169795000x6967; fax: +1 4169795336. E-mail address: [email protected] On leave from the Department of Mathematics, Ryerson University, Toronto, Ont., Canada.
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.052
ð1Þ
408
G.N. Ord / Chaos, Solitons and Fractals 30 (2006) 407–411
This is followed by the relativistic requirement that E2 ¼ m2 þ p2 ;
ð2Þ
where m is the rest mass of the electron and p is the momentum. Combining these requirements lead Dirac to ih
ow ¼ ða p þ bmÞw; ot
ð3Þ
with suitable anticommutation relations for the matrices a and b. Much as the original argument was brilliant and insightful at the time, there was no sense in which the resulting equation described a wavefunction w that had a prior physical meaning. As in the non-relativistic case, w here is an abstract tool in a probability calculus. The objective of the calculus is to obtain expectation values of ‘observables’ using w. Whether w has any physical analog in Nature is unknown to this day. Now solutions of (3) involve wave propagation, spin, the uncertainty principle, special relativity, and a host of features we commonly associate with relativistic quantum theory. It is natural to assume then that the equation itself is intimately related to some pretty complicated physics. How then are we to extract such an equation from a simple accounting problem? The trick is to look beneath the elegant mathematical structure of the differential equation for a simple statistical mechanical picture.2 With such a picture in place we shall see that we can indeed understand the equation in terms of simple accountancy tools. The added bonus will be that as accountants, we shall always know what we are counting; we shall know what w represents.
2. The Entwined path model We consider a special ‘‘entwined’’ walk on a square lattice in the (x, y)-plane [8–10]. The lattice spacing will be some small positive rational number and the walker will move at each step in one of the four directions (±1, ±1). The stochastic element of the walk is governed by the binomial distribution. That is, at step (n + 1), the walker will step in the same direction as in step n with probability (1 m), with m a fixed positive constant and m 1. The alternative at each step, besides continuing in the same direction, is a change of state. A change of state occurs in an alternating pattern. The walk is started from the origin and the first step is to (1, 1). The stochastic process is consulted and the walk continues to (2, 2) unless a state change is indicated. The first state change is a change in the direction of motion along the x-axis. So if this happens at (1, 1), the subsequent step is to (0, 2). The next state change will not change the direction of the walk, but the walker will drop a marker for its return path before stepping to the next site. Thus all the odd numbered state changes will be direction changes in the x coordinate, all the even numbered state changes will preserve direction but drop markers. This process will continue until the walk steps past some pre-assigned y-coordinate, yM say, at which point, at the next marker drop, the walker will return to the origin along the path defined by the set of markers. A single such entwined path (EP) is illustrated in Fig. 1. Notice that the EP defines a chain of oriented rectangles in which the orientation reverses at each crossing point of the EP. If we use the ‘right hand rule’ for path traversal, the first rectangle has positive orientation out of the page, the second negative, into the page, and so on in an alternating fashion. As the walk is lengthened by repeating the process we can imagine that a cone with apex at the origin will become covered by an ensemble of these oriented rectangles. If we sit at a particular off-lattice point inside this cone we could expect to be regularly encircled by these oriented rectangles and we might expect that we should eventually see an equilibrium distribution of orientation, varying from point to point throughout the cone. How could we measure this orientation? Consider again Fig. 1. Each oriented rectangle has a left and right boundary that contributes to the oriented area. So as to avoid counting the contributions of an area twice let us restrict our attention to the right boundary of each rectangle. The collection of right boundaries itself forms a path that we call the ‘enumerative path’. Note that the enumerative path has a very regular structure and we can use it to count oriented rectangles. The blue segments of the path belong to positively oriented rectangles and add a + 1 to the oriented area count. The red portions of the enumerative path belong to negatively oriented rectangles and contribute a weight of 1 to the count. Now our task to count oriented rectangles is reduced to counting the contributions of enumerative paths on the lattice. We can do this using simple combinatorics, or we can assume an equilibrium is reached and use the structure of the walks to deduce what the equilibrium pattern must be. The latter route is the simplest.
2
This idea was initiated by Feynman in what became known as his ‘Chessboard model’ [1–7].
G.N. Ord / Chaos, Solitons and Fractals 30 (2006) 407–411
409
+
–
+
–
+
Fig. 1. On the left is a single ‘entwined path’ in the (x, y) plane. The colour indicates the direction of traversal, blue for traversal in the +y direction, red for traversal in the y direction. The origin is at the bottom of the path and the lattice spacing is small on the scale of the figure. Notice that the crossing of the forward (blue) and backward (red) paths forms a chain of oriented areas. The orientation switches from one rectangular area to the next. This is illustrated by the recolouring of the path in the centre figure. There blue represents positive orientation and red represents negative orientation. To count oriented rectangles we can simply use the ‘enumerative’ path, illustrated on the right hand side of the figure. Here the colouring illustrates the contribution to orientation, +1 for blue, 1 for red. (For interpretation of the references in colour in this figure caption, the reader is referred to the web version of this article.)
Let us label the lattice sites by x = m, and y = n where m and n are integers. We consider a two-component density u±(x, y) where u+ counts the number of (1, 1) and (1, 1) directed links and u counts the number of (1, 1) and (1, 1) directed links by orientation. We need a two-component density here because our enumerative paths continually shuffle orientation counts between the two directions. Now any link at (x, y + ) in the ±(1, 1) direction either follows a link of the same direction and colour at (x , y) or follows a link of the opposite direction and colour at (x + , y). The former occurs with probability 1 m, the latter with probability m. Thus if an equilibrium density is reached it must satisfy uþ ðx; y þ Þ ¼ ð1 mÞuþ ðx ; yÞ mu ðx þ ; yÞ.
ð4Þ
Notice here the subtraction involved in the second term. This is because whenever our enumerative path ‘turns right’ it switches orientation, thus changing the sign of its contribution. Since this happens for all paths, it must happen for the equilibrium distribution. We can similarly deduce that the u density must obey the difference equation u ðx; y þ Þ ¼ ð1 mÞu ðx þ ; yÞ þ muþ ðx ; yÞ.
ð5Þ
The positive sign for the second term reflects the fact that the change of direction for a ‘left turn’ on an enumerative path does not change orientation. Eqs. (4) and (5) are difference equations that, with suitable initial conditions for the u± yield rational number solutions inside the cone above the x-axis. The equations themselves are just conservation equations that express the fact that the enumerative paths are continuous and have the alternating orientation of Fig. 1. The solutions u± spread out into ever larger regions along the x-axis as y increases, so u includes an exponential decay. We may choose to look through this background decay to see the emerging pattern by following the evolution of: w ðx; nÞ ¼ u ðx; nÞð1 mÞn ;
ð6Þ
which satisfies the equations wþ ðx; y þ Þ ¼ wþ ðx ; yÞ mw ðx þ ; yÞ; w ðx; y þ Þ ¼ w ðx þ ; yÞ þ mwþ ðx ; yÞ to lowest order in .
ð7Þ
410
G.N. Ord / Chaos, Solitons and Fractals 30 (2006) 407–411
Let us now pause and note that Eq. (7) is a perfectly respectable equation for an accountant to produce. There are no abstractions or mathematical tools involved that take us out of the domain of good accountancy. We know exactly what is being counted . . . oriented rectangles produced by a long entwined path. We know the number system involved . . . the counting actually just used integers but the use of probability and normalization placed us in the domain of rational numbers. We have assumed that an equilibrium pattern will emerge from our initial conditions but that is something that is easily checked by performing a numerical experiment. Indeed, the calculation up to this point involves only a basic counting of recognizable objects on a planar lattice. Now physicists are generally impatient with difference equations and restrictions to rational numbers. So now let us be physicists once more and do something an accountant is unlikely to do, namely idealize the situation by taking a continuum limit. If we subtract w±(x, y) from both sides of (7), divide by and take the limit as ! 0+ we find that oy wþ ðx; yÞ ¼ ox wþ ðx; yÞ mw ðx; yÞ; oy w ðx; yÞ ¼ ox w ðx; yÞ þ mwþ ðx; yÞ 1 0 0 1 wþ ; rz ¼ and rq ¼ ¼ iry we have or writing w ¼ w 0 1 1 0 oy w ¼ rz ox w imry w.
ð8Þ
ð9Þ
This may be recognized as a form of the Dirac equation where c = h = 1 and y = t. Note that if we iterate this equation to get a second order form we have o2y w ¼ o2x w m2 w;
ð10Þ
which is the Klein-Gordon equation. Notice that to get (9) and (10) we have not invoked: (A) (B) (C) (D) (E)
the uncertainty principle, quantization or Schro¨dinger’s equation, complex numbers, special relativity, any ‘interpretation’ regarding quantum mechanics and the nature of reality.
We have simply taken the output of an ‘accounting argument’ and written it in a language familiar in the context of relativistic quantum mechanics. Now (9) is just a continuum limit of (7) written in a familiar form. (Notice that there has been no analytic continuation forced on the system, w is real and the i in (9) is present only because ry is imaginary.) The point here is that we can regard (9) either as a fundamental equation about the ‘wavefunction’ of an electron, without knowing exactly what a wavefunction represents in the physical world, or we can take (9) as the continuum limit of an equation describing an equilibrium distribution of a simple stochastic process. The continuum language that we use does not tell us whether we are describing a ‘Dirac wavefunction in one dimension’ or a ‘spacetime that maintains an accountancy ledger for the EP stochastic process’. Since Dirac wavefunctions are rather mysterious objects, it is worthwhile going through the above list to see how each property is manifest in the stochastic model. Here is a brief summary. (A) The uncertainty principle. This is built into the accountancy model by the fractal geometry of the EP’s themselves. If we look at a single enumerative path on scales much greater than 1/m we will see that it scales as a random walk in x, giving a fractal dimension of 2. This gives the correct scaling for the uncertainty principle. (B/C) Oscillatory solutions. The form of the accountancy problem solutions arises from the fact that we are counting oriented areas. Had we not recognized orientation in the counting, the minus sign on the right in (4) would have been a plus sign and the matrix iry in (9) would be replaced by rx. With that replacement, both (9) and (10) would become forms of the telegraph equations, which describe diffusive processes. Thus the ‘quantum’ aspect of oscillatory solutions and the implication of complex numbers all come from the counting of oriented areas, signaled by the presence of the matrix rq. Although this matrix is real, it has eigenvalues ±i which account for the oscillatory nature of the solutions. Dx (D) Special relativity. The paths in the stochastic model move on the diagonal so that at each step Dy ¼ 1. This translates to Dx ¼ 1 in the Dirac equation. Thus the constancy of the speed of light c = 1 is respected in the Dt model. The interpretation of a ‘Lorentz transformation’ is more subtle, involving both the geometry of the paths and the algebraic properties induced by orientation.
G.N. Ord / Chaos, Solitons and Fractals 30 (2006) 407–411
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(E) Interpretations. The accountancy problem received no input from any particular view of a ‘quantum mechanical reality’. When we set up the problem, we did so assuming a classical distinction between object (the EP) and the observer (the accountant). There was no sense in which these were mixed. The random walker left a trail on the (x, y)-plane and the accountant simply counted links in the path by orientation. We did not have to discuss a ‘state of knowledge’ of the accountant, nor was the walker obliged to interact with the accountant. The wavefunction solutions of the difference equation had no apparent dependence on, or connection to the accountant’s consciousness, or for that matter to multiple universes, pilot waves or ensembles of similar systems. In the accounting problem the wavefunction was a manifestation of what J.S. Bell could have called a ‘beable’ . . . the EP. Since the wavefunction in the accounting calculation has an unambiguous interpretation, this provides an illustration that the problems of interpretation in quantum mechanics stem not from the unitary evolution of the wavefunction, but from the measurement postulates. Given that it is generally understood that measurement, not propagation is the prime difficulty for interpretations of quantum mechanics, is there something to be gained from this demonstration? I think the answer is yes! Notice that to see that there is an interpretation of Eq. (9) that can be explained in simple terms to an accountant, we have to look at the dynamical process and show how the act of counting oriented areas leads, in equilibrium, to a difference equation that may in turn be approximated by a differential equation. We had to see how counting, equilibrium and a continuum limit all conspired to give us a compact language to describe the final result. Had we started with solutions of the Dirac equation and been asked to interpret them for an accountant, we would have been hard pressed to do so. The ‘natural’ interpretation of solutions of the Dirac equation is in terms of waves. These are not features of entwined paths themselves but objects constructed by them. By comparison, in the interpretation of quantum mechanics we usually take wavefunction behaviour as a starting point for discussions of experiments on real systems. The accountancy problem shows that this may limit our view of what is happening in nature. Our mathematical language that works well for the calculation of statistical averages may be less than effective at revealing processes underlying those averages.
Acknowledgement The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada.
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