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On a New Mode of Description in Physics.
David Bohm and Basil J. Hiley
Department of Physics, Birkbeck College, University of London
and
Allan E. G. Stuart
Department of Mathematics, The City University, London
Received: 15 December 1969
[Appeared in Int. J. Theor. Phys. 3, 171-183, (1970)]
Abstract
We explore the possibilities of a new informal language, applicable
to the microdomain, which enables such characteristics as superposi-
tion and discreteness to be introduced without recourse to the quantum
algorithm. In terms of new notions that are introduced (e.g. ‘potentia-
tion’ and ‘ensemblation’), we show that an experiment need no longer
be thought of as a procedure designed to investigate a property of
a ‘separately existing system’. Thus, the necessity of a sharp sepa-
ration between the ‘system under observation’ and the ‘apparatus’ is
avoided. Although the new language is very different from that of
classical physics, classical notions appear as a special limiting case.
This new informal language leads to a mathematical formalism which
employs the descriptive terms of a cohomology theory with values in
the integers. Thus our theory is not based on the use of a space-
time description, continuous or otherwise. In the appropriate limit,
the mathematical formalism contains certain features similar to those
of classical field theories. It is therefore suggested that all the field
equations of physics can be re-expressed in terms of our theory in a
way that is independent of their space- time description. This point
is illustrated by Maxwell’s equations, which are understood in terms
of cohomology on a discrete complex. In this description, the electro-
magnetic four-vector potential and the four-current can be discussed in
terms of an ‘ensemblation’ of discontinuous hypersurfaces or varieties.
Since the cohomology is defined on the integers the charge is naturally
discrete.
1
1 Introduction.
Various forms of contemporary relativistic quantum field theories and the
closely related S-matrix approach make the explicit assumption that all the
physically relevant properties may be obtained by ordering the fields or
scattering amplitudes on four-dimensional differential manifolds. The suc-
cess that has been achieved using this assumption, particularly, for example,
in quantum electrodynamics, has created the feeling that these theories or,
at least, closely related theories are essentially correct. This has led to a
considerable effort to search for better mathematical techniques with which
to overcome the outstanding problems.
We recognise the value of the work done along these lines, but we wish to
point out that a description based on a differential manifold has serious limi-
tations. Indeed, various attempts havc already been made to investigatc the
precise nature of these limitations (Atkinson and Halpern, 1967; Hill, 1955;
Schild, 1949; Snyder, 1947). Some of these enquiries have considered in what
way non-usual topologies affect the results, while others have attempted to
remove the need for renormalization by adopting a discrete topology, usually
by introducing some form of fundamental length. However, these attempts
mainly investigate the effects of various assump-tions on the formal aspects
of the theory. Indeed, there is a widespread feeling that a change in the
informal language (i.e. that used to describe the experimental situation) is
neither necessary nor even possible (Bohr, 1934, 1958). This means that,
once again, there is a heavy emphasis on thc mathematics.
It is our view that it is not sufficient to change only thc formaJism
and that, as has already been argued elsewhere, a more radical approach
is needed (Bohm, 1968; Hiley, 1968). New informal languages and their
extensions into mathematical forms need to bc investigated.
In any new approach that is relevant to physics, discreteness should
appear as a natural consequence of the informal considerations and should
not be arbitrarily imposed. Thus it is not possible to obtain discreteness
naturally if the classical notions of particie, trajectory, potential, field, etc.,
continue to be taken as primitive concepts since these notions were developed
specifically for the continuum. Of course, it is necessary for such notions to
emerge at some more abstract level as a result of, say, some form of suitable
statistical averaging procedure. However, in this way the classical forms will
arise as a consequence of some deeper, more primitive theory.
In this paper we consider a radically new theory that uses a novel infor-
mal language which is very different from that usually adopted in physics
today. In contrast to our present way of using language, which places empha-
2
sis on separate objects in interaction, we give primary relevance to activity
and wholeness in the sense of undivided movement.
Our main reason for emphasising these notions is that they are implicit
in quantum theory. For example, the indivisibility of the quantum of action
implies a merging of the ‘observed system’ and the ‘observing apparatus’
so that the two are inseparable and, therefore, constitute a whole in which
analysis into parts is not relevant. This whole flows and merges into the
totality of the universe, including the human observer.
In exploring new informal languages, wc have been guided by our ex-
perience with quantum theory. For instance, each ‘quantum state’ implies
potentialities (Bohm, 1960) whose realisation can be incompatible in the
sense that different realisations require mutually exclusive experi-mental ar-
rangements. In a description which uses the informal language of classical
physics, this incompatibility is understood in terms of the Principle of Com-
plementarity. In our new informal language we do not need any such princi-
ple. However. we do need new terms in order to call attention to a different
way of thinking. We thus introduce a ncw basic term ‘potentiation’. The
full meaning of this term will be discussed in Sections 2 and 3 but, for the
present, we can regard it as meaning the realisation of potentialities.
In terms of potentiation, an experiment has a new meaning (see Section
3) In classical physics the results always refer to the properties of a sys-
tem existing separately from the observing apparatus. In our description
there is no separately existing system. The overall experimental arrange-
ment potentiates a content, the meaning of which depends explicitly on this
arrangement. Thus there is no separation of a ‘system’ under observation
from an ‘observing apparatus’; and this is just what is implied by the finite
nature of the quantum of action.
Further, from quantum theory, we know that the potentialities are not
discussed in terms of individual events but in terms of ensemble averages.
Again, in our new informal language we introduce another basic term,
namely, ‘ensemblation’ which essentially means the formation of ensembles.
As will be seen in Section 2, these ensembles are not only of the type en-
countered in statistical mechanics but can also be of a very different nature.
Thus the essential features of the quantum theory are contained in the
notions of discreteness, potentiation and ensemblation. These features arise
naturally in a certain mathematical description which makes use of the de-
scriptive terms of a cohomology theory with values in the integers (Cairns,
1958). This theory also contains a linear superposition principle analogous
to that of the quantum theory (see Section 3). Our superposition principle
differs from that of the quantum theory, however, in the sense that it cannot
3
be expressed in tcrms of a Hilbert space (except as an approximation valid
in a suitable limiting case). Indeed, it is because of this that we can describe
the experiment in a different way which does not give fundamental relevance
to the Principle of Complementarity or to the Uncertainty Principle.
As a matter offact, we show that in terms of our new mode of description,
the ‘wholeness’ of the ‘observing instrument’ and the ‘observed content’ is
just as relevant in the classical domain as it is in the context of quantum
theory. Thus, to illustrate, we consider as a particular example, the expres-
sion of the laws of classical elcctrodynamics in terms of cohomology theory.
Indeed, Misner and Wheeler (1957) have already indicated that once the
differential equations are written in terms of differential forms and exterior
derivatives, these equations can be re-interpreted as defining a de Rham
cohomology. In this way, they have been led to propose that (continuously
variable) charge can be explained as a certain topological aspect of space-
time, i.e. a ‘worm hole’ in which the electromagnetic field is ‘trapped’. We
suggest instead, however, that the de Rham cohomology can be understood,
in the light of our theory, as defining a cohomology on abstractsimplicial
complexes with values in the integers. In this way, the equations can be given
a meaning that is independent of whether or not there is an underlying space-
time description, and of whether the structures involved are con-tinuous or
discrete. In our illustration, in terms of the laws of classical electrodynamics,
we are thus led to propose a new meaning for the electro-magnetic four-
potential and for the four-current, in which charge is naturally discrete.
The detailed contents of this paper are as follows. In Section 2 we in-
troduce and discuss the new basic notions, while in Section 3 we propose a
mathematical description of potentiation and ensemblation. In Section 4 we
consider as a particular example the expression of the laws of classical elec-
trodynamics in terms of cohomology theory, and show how our description
leads to a discrete charge.
2 Informal and Formal Languages in Physics.
Before the development of the quantum theory, the fundamental descriptive
language of physics contained the assumption, either implicit or explicit,
that the world is constituted of separately existing objects, i.e., ‘things in
themselves’, which interact with each other according to well-defined laws.
Bohr was probably the first to realise that the finite nature of the quantum
of action implies that this description cannot be relevant in the quantum
context. Thus the separation of the ‘observed object’ from the ‘observing
4
apparatus’ is no longer a tenable form of description and this means that
‘observed properties’ cannot consistently be attributed to the ‘object’ alone.
Originally, it was implied that this situation is the outcome of an unknown
disturbance of the ‘observed system’ in its interaction with the ‘observing
apparatus’. But, as was pointed out, especially by Bohr, the implications of
the quantum of action go much further than this, and indeed call into ques-
tion the entire notion of an ‘object in itself’ as a relevant form of description
of physics.
However, along with most other physicists, Bohr maintained that or-
dinary common-sense language, refined where necessary with the language
and the concepts of classical physics, has to be used in the description of the
experiment, and that any other form of description of the experiment is im-
possible. In this way, there was established a sharp separation between the
language for describing experiments and the formal mathematical language
used for making theoretical inferences about the results of the experiment.
To be sure, a certain relationship of correspondence between these two kinds
of language was indicated. But the formal mathematical terms (i.e. Hilbert
space, operators, commutators, etc.) were regarded as having no relevance
for discussing the experiment itself.
The informal language relating theory and experiment (i.e., probabilities,
scattering amplitudes, etc.) was thus regarded as essentially determined and
unchangeable, even though the formalisms, of course, underwent quite con-
siderable alterations (e.g. renormalisation, S-matrices, Regge pole theory,
etc.). Indeed, there appears to have arisen a widespread tacit agreement
that basic advances would now have to be made in the mathe- matics alone,
while the general informal language relating the experiment to theory would
continue, more or less as it had been since it was originally introduced in
connection with Schr¨odinger’s equation.
This approach still, however, treats the results of an experiment as an
‘object of discourse’ which can be abstracted from the overall experimen-
tal arrangements and made the subject of mathematical inferences, which
ultimately refer to ‘laws of nature’ that would be independent of, and sep-
arate from, the apparatus. In a sense, therefore, the ‘thing in itself ’ is still
relevant but at some more abstract level. However, to hold onto the ‘thing
in itself’ in this way is still not consistent with the full implications of the
quantum of action. To go beyond the customary mode of description which
leads to this inconsistency, we have to question the assumption that ordi-
nary common-sense language (refined with the aid of classical concepts) is
the only possible one for discussing the experiment.
In particular, it is necessary to go outside of what is generally regarded
5
as the domain of physics and to enquire into our perceptions, from which
our knowledge of objects must eventually come. Is the notion of an object
actually basic in perception, or is it that we have come to regard such a
notion as self-evident, and therefore fundamental, because of environ-mental
conditioning and training?
There is a great deal of experimental evidence coming from psychologi-
cal and neurological investigations that, beginning from early childhood, we
actually learn to abstract the notion of an object from a more fundamental
level of perception (Bohm, 1965b). What is primitive is perception of move-
ment, or of change, or of a break in some regular order or arrangement. From
an ensemble of such perceptions of movement, something relatively invariant
is abstracted, and this abstraction is the foundation for the prese,ntation of
perception in the form of relatively fixed or slowly moving objects. This
is indeed very similar to what happens in relativity in which, likewise, the
‘object’ is abstracted from invariants of movement (Bohm, 1965a).
We are thus led to suggest that primitive perception is close, in a cer-
tain sense, to the most advanced developments of physics, whereas classical
physics, or ‘common-sense’ descriptions, are high-level abstractions which
we have learned to regard as fundamental because of an extensive process
of conditioning. Thus in primitive perceptions, the ‘thing in itself ’ is not
fundamentally relevant and the same holds in quantum theory and in rela-
tivity.
However, because our general language has been developed to meet cer-
tain everyday needs with regard to the use of objects, the noun, which is the
indication of such an object, has been given a fundamental role, while the
verb, which calls attention to action, tends to have a secondary importance.
Therefore, to cease to take the ‘thing in itself’ as primitive, we will instead
give a basic role to the verb (while nouns will be regarded as abstrac-tions
from verbs). This approach emphasises movement and activity and implies
that objects are, even in informal discourse, to be regarded as rela-tive in-
variants of such movement and activity. Thus, the ‘object in itself’ is no
longer taken as a basic term of description. Since movements can gener-
ally flow into each other and merge, the division of the world into separate
constituents has also been dropped. Like the ‘object’, it will arise as an
abstraction from an unbroken and undivided totality of movement, which
we call holomovement (using the Greek prefix ‘holo’, which means ‘whole’).
In a given context, certain aspects of such a totality will be relevant,
while others will not. It is useful here to bring back the word ‘to relevate’,
which has dropped out of common usage. This means ‘to lift into attention
(e.g. as in ‘relief’). In any perception, certain aspects can thus be said to
6
be relevated.
We may extend the usage of this word to say that, in a certain sense,
a given experimental arrangement also relevates a content. In accordance
with our discussion in the introduction, this arrangement actively helps to
create conditions for a particular phenomenon to appear in a particular
form and thus to stand out as being relevant. By using the verbal form, ‘to
potentiate’, we emphasise this active role (and, of course, we cease to regard
the ‘thing in itself’ as a fundamental descriptive term).
However, we do not wish to imply that only an experimental apparatus
can relevate and potentiate a content. On the contrary, we propose that
any arrangement of matter potentiates and relevates a certain content, and
that the action of the apparatus is thus a special case of this, in which the
outcome is particularly simple to interpret. Since the observing apparatus
is not given a special role in the description, it follows that the subjective
observer also has no special role.
As pointed out in the introduction, the potentiated content can generally
be described as an ensemblation. In this connection, it is significant to note
that the definition of ‘ensemble’ given in the dictionary is that each member
is related only to a whole. This feature of an ensemble can be illustrated
by considering a painting. The individual spots of paint can be said to
ensem- blate, to form a whole content, including trees, houses, etc. This
whole content is evidently of a very different character from the individual
spots of paint, whose only significant relation is that they form such a whole.
Similarly, movements ensemblate to form wholes. Thus in perception, all
the changes or breaks in movcment ensemblate to give rise to the rela- tively
invariant objects.
In physics, we not only have statistical ensemblations but also ensembla-
tions of a more general kind. For example, in bubble chambers, we see a
sequence of dots which form an irregular curve that is interpreted as the
track of a particle. This irregular curve can be described as an ensemblation
of a certain number of bubbles. In addition, these bubbles may be further
ensemblated to a ‘smooth curve’, which is a sort of average ‘track’ and, in
turn, a number of these ‘tracks’ ensemblate to form what may be called
‘a whole picture’, describable mathematically in terms of wave functions or
‘quantum states’. Such wave functions are as different in character from the
bubbles as the content of a painting is from the spots of paint.
As indicated earlier, the further developments of the mathematical de-
scription of the ‘whole picture’ (e.g. field theories, S-matrices, etc.) are
limited by the ‘classical’ informal language currently used for describing the
experiment. Because our own informal language, as discussed above, is dif-
7
ferent in that it emphasises movement and wholeness, we can introduce new
forms of mathematics going outside the limits of theories that can be inter-
preted in terms of a ‘classical’ informal description. Thus, for example, our
descriptions in tcrms of simplicial complexes, using a cohomology over the
integers (see next section), goes beyond the notion of a Hilbert space. The
entire scheme of unitary transformations with its probability inter-pretation
is no longer fundamentally relevant (except in suitable limited cases). More-
over, quite new directions of enquiry are opened up, which we shall explore
in later papers.
3 Basic Mathematical Description of Ensembla-
tion.
In the previous section we have introduced some new informal terms, i.e.
potentiation and ensemblation, which are relevant in physics. We now in-
troduce a mathematical formalism in which the potentiations and ensem-
blations can be given a more articulated and detailed description.
The very notion of potentiation is essentially non-local so that a mathe-
matical theory using continuous coordinates is inappropriate (as will become
apparent when we attempt to relate our theory to the conventional ones).
Fortunately in combinatorial topology, a mathematical theory which does
not depend on locality has already been developed and appears very suit-
able for the detailed description of our ensemblations. This is the theory of
homology and cohomology (Hilton and Wylie, 1960).
We use the terms of homology theory as basic descriptive forms. We
begin by calling each potentiation an abstract 0-simplex. Then if two po-
tentiations, Aand B, are related in some way, we will represent this relation
by a ‘connection’ or ‘line’ and call the relation A◦B, an abstract 1-simplex.
In our later work we will find it convenient to distinguish between the re-
lations A◦Band B◦A. This can be achieved by attaching a ‘direction’
or ‘orientation’ to the abstract simplex. A cyclic relationship between three
potentiations A◦B◦Cwill be represented by an abstract oriented 2-simplex
and so on. (For convenience, we will simply use the term simplex and un-
derstand it to mean an abstract oriented simplex.) Thus the potentiations
and their relationships are said to form a simplicial complex which can be
used to give a detailed description of a totality of potentiations.
We can form an ensemblation from such potentiations in a variety of
ways. For example, we can relevate the various potentiations by weighting
them with suitable integer coefficients. Or we can relevate the various rela-
8
tionships between different potentiations by a similar weighting of simplexes
of higher dimension. When the integers are thus used as weights, the rel-
evant ensemblations are equivalent to what are called in topology, integral
chains on the complex. The dimensionality of the chains will depend on
whether we are considering the 0-simplexes, the 1-simplexes, etc. (e.g. a
1-chain will be a linear combination of 1-simplexes suitably weighted).
In this ensemblation we can now use what are called the boundary op-
erators to find the subset of chains which are cycles (i.e. which have no
boundary) and the subset of cycles which are bounding cycles (i.e. those
cycles which are boundaries of chains). In this way we can discuss the ho-
mology properties of the ensemblation.
We propose to describe the results of an experiment as an ensemblation
of chains and cycles on a simplicial complex of potentiations. Thus the
results are actively potentiated in the holomovement and are not treated as
‘things in themselves’. The holomovement, however, involves the apparatus
and the general background, along with the results, in an inseparable way.
And, as indicated in earlier sections, this implies that we have to use the
same informal and formal languages to describe the apparatus as we use for
discussing the results of experiments, along with the inferences to be drawn
from these results.
The form and structure, as well as the activity of the apparatus can be
described in terms of a dual complex which we shall call the complex of
copotentiations. What is relevant for the theoretical inferences is a certain
relationship of potentiations and copotentiations which is mathematically
termed their ‘intersection’ and which is invariant to the changes of the basis
of the simplicial description. Thus, as far as the theory is concerned, the
‘common sense’ language of classical physics no longer plays a fundamental
role in the description.
Of course, the distinction between a potentiation and copotentiation is
merely a convenient form of description and is not to be taken as implying
their separate existence. Indeed, what are taken as copotentiations at one
level can be regarded as potentiations on another level (e.g. the copotenti-
ations describing the apparatus can be expressed as potentiations on a ‘finer
mesh’). Thus ultimately our approach implies a hierarchy of complexes,
though in any given context a limited nuro ber of steps in the hierarchy will
be adequate.
If we regard the dual of a complex as a functional, we can use the descrip-
tive terms of cohomology theory for the copotentiations. By using integral
weights for the copotentiations and for the relationships between them, we
obtain the corresponding cochains and cocycles.
9
The ensemblations, which are to be regarded as the physically relevant
content, will (as has already been pointed out) be described in terms of the
intersection of the relevant chains and cycles with the corresponding cochains
and cocycles. That is, if the relevant physical situation is described by sets
of p-chains, Cj
(p)and by sets of p-cochains, C(p)
k, then the intersection is
written as
(C(p)
k, Cj
(p)) = gj
k(1)
where gj
kis an integer. In fact, each situation will be described by matrices
with integer elements which will be called intersection matrices. Thus our
theory contains a type of discreteness which seems to be called for in quan-
tum theory and in other physical theories (e.g. discrete charge in electro-
dynamics).
A further important aspect of this general type of description is that
it contains a superposition principle analogous to the one used in quantum
theory. This is because the chains and cycles, as weIl as the cochains and
cocycles, can be added with integer coefficients to form an Abelian group.
Or, to put it less formally, given any two physicaIly relevant chains C1
(p)and
C2
(p), then if αand βare integers, the chain
C3
(p)=αC1
(p)+βC 2
(p)(2)
is also a physically relevant chain. This is evidently similar to thesupcr-
position principles that are used not only in quantum theory, but also in
other branches of physics. The key difference is that the coefficients are
restricted to integers (so that, for example, we are not dealing with a Hilbert
space). However, in the limit when the integers are very large, it is clear that
our chains correspond in some approximate sense to vectors and tensors in
Hilbert space.
Because there is a physical meaning to linear superposition, we can now
define linear transformations of the chains and cochains which are similar to
the unitary transformations in quantum theory but which differ in important
ways as well. Thus a linear transformation on the p-chains can be written
as
C0j
(p)=aj
iCi
(p)(3)
while the corresponding adjoint transformations on the cochains is
C0(p)
j=C(p)
jbm
k(4)
10
The intersection matrix then transforms bilinearly in the foIlowing way
g0=agb (5)
A further analogy with quantum theory is now evident. For the ‘observables’
of quantum theory, which are represented by matrices Aij , undergo a similar
bilinear transformation under change of basis
A0=S†AS (6)
In our theory, the physicaIly relevant ‘observable’ will, of course, be de-
scribed in terms of intersection matrices rather than in terms of usual quan-
tum mechanical matrices which represent operators in a Hilbert space.
Once the implications of our approach have been understood, it is not
just a simple matter of looking at the various experiments that have al-
ready been performed and trying to understand them in terms of the present
scheme. Many experiments are designed to test specific questions as raised
in the particular form used in a given theory. For example, in classical
physics there are questions about particle orbits, causal ordering by means
of fields, etc., while in the quantum domain there are questions about energy
levels and about probabilities of various processes. Our new description is
radically different and thercfore the old experiments may not ask questions
that are appropriate to it.
The descriptive form that we have introduced has to be taken further be-
fore we can understand exactly how a particular apparatus can be described
in terms of a particular set of copotentiations. We leave this question on one
side and will take it up again in a later paper. However, in the next section
of the present paper we shall consider an illustrative classical example in
which a similar question arises, but in a simpler way.
4 The Laws of Electrodynamics Described in Terms
of Our New Language.
The laws of electrodynamics were first expresscd in terms of integrals of
fields over cycles of varying dimensionality, e.g. Ampere’s law, Faraday’s
law, Gauss’s law, etc. It is only from the extrapolation of these integral laws
to infinitely small cycles that one obtains Maxwell’s equations. Thus these
equations go considerably beyond what can be inferrcd from observations
alone. The relative ease of the mathematical application of thc differential
form of Maxwell’s equations has made this approach attractive. However,
11
the infinities which arise in the indefinite extension of this form, both clas-
sically and quantum mechanically, imply that it may be appro-priate to go
back to the integral form in spite of the possibility of greater mathematical
difficulty. The appropriate mathematics for doing this is just the theory of
complexes of chains and cochains that we have described earlier.
In this paper, we will restrict ourselves to classical electrodynamics. In
our description, the relevance of the wholeness of the instruments and the
observed content is evident even in the classical context. For every observa-
tion is the integral of a field quantity over a cycle. The field is a potentiality
of ‘empty space’. Moreover, the cycles are also potentialities in the sense
that the cycle which is physically relevant in any given situation will depend
on the experimental arrangement. Remembering that the role of the chain
and cochain is conventional, so that the two can be interchanged, we find
it convenient to describe the fields in terms of cochains, while the cycles
are described in terms of chains. The intersection matrices of chains and
cochains (which in ordinary terminology would be the integral of fields over
cycles) is then the physically relevant quantity in terms of which the laws of
physics are to be expressed. We are thus led to a kind of ‘wholeness’ very
similar to that arising in the context of quantum theory.
In order to help visualise what is meant, we will describe the fields and
the cycles in terms of the usual space-time description involving vectors,
tensors, etc. (but recalling that this is a particular case of our general
description in terms of chains and cochains). We begin with the vector
potential AµUsually this is taken as a continuous function but we are going
to regard it as a discontinuous one which resembles a set of δ-functions in
the sense ZC(1)
Aµdxµ=n(7)
where nis an integer. One may visualise this by thinking that in any given
region there is an array of surfaces, e.g., hyperplanes, at which the vector
potential undergoes a discontinuous change such that as the 1-chain, C(1),
crosses the surface, the integral increases by unity. In other words, ncounts
the number of planes crossed by C(1). Let us now consider the meaning of
Stokes’ theorem,
ZBC(2)
Aµdxµ=ZC(2)
∂νAµdxν∧dxµ(8)
where C(2) is a two-dimensional area and BC(2) is, by definition, the bound-
ary of this area. If one thinks of the meaning of the right-hand side, one
12
can see that the integral can fail to be zero only if some of the surfaces cor-
responding to Aµhave boundaries which cross the area C(2). To make the
example yet simpler, consider the projection of the vectors and tensors into
a three-dimensional sub-space; then the meaning of Stokes’ theorem is that
the number of surfaces having boundary lines which cross C(2) is equal to
the number of surfaces crossed by the boundary of C(2). This means that in
a certain sense it may be said that ∂ν∧Aµcorresponds to the boundary of
the surfaces described by Aµ. This kind of correspondence can be extended
to any number of dimensions.
As an example, the electromagnetic field tensor, Fµν will correspond
to a two-dimensional variety and ∂α∧Fµν corresponds to the boundary
of this variety which is one-dimensional. Thus Maxwell’s first equation,
∂α∧Fµν = 0, implies that the variety corresponding to Fµν is closed, i.e., it
has no boundary. If the topology is ‘trivial’ this means that Fµν =∂ν∧Aµ
Or, in other words, the Fµν variety is the boundary of the hyperplanes
corresponding to Aµ. If we restrict ourselves to a three-dimensional sub-
space, the Fµν will correspond to discrete lines of force for the fields. In the
limit, where the varieties are ‘very dense’, we will get a quasi-continuous
description.
We now put the abovc results in terms of our own language of chains and
cochains so that the laws will be expressed independently of any underlying
continuous space-time. The basic physically relevant quantity RAFµνdxµ∧
dxνcorresponding to the integral of the field over the area Ais now to be
written as
(f(2), C(2) ) = n(9)
where nis an integer, C(2) is a 2-chain corresponding to the cycle of inte-
gration and f(2) is the 2-cochain corresponding to the field. Maxwell’s first
equation,
ZC(3)
∂αFµν dxα∧dxµ∧dxν=ZBC(3)
Fµν dxµ∧dxν= 0 (10)
now corresponds to
(Bf (2), C(3) ) = (f(2), B C(3)) = 0 (11)
Here Bis the boundary operator. The meaning, for example, of BC(2) is
that one is to replace the 2-chain C(2) by that 1-chain which is its boundary.
Thus we obtain laws in which differential operators are no longer used and
in which assumptions of continuity are not needed.
13
Maxwell’s second equation requires the definition of the dual tensor
F∗
µν . In continuous geometry this is done with the aid of a metric F∗
µν =
1
2µναβ gαλ gβκ Fλκ. The dual to any element is essentially a kind of perpen-
dicular or normal. Thus the dual to Aµis µλgλν Aνwhich corresponds to
an ensemble of lines that is perpendicular to the surfaces defined by Aµ. In
terms of topological notions, however, there is no meaning to perpendicu-
larity. Rather, the dual has to be defined in a different way which does not
require the metric tensor in any basic sense. In later papers we shall discuss
how this is to be done. However, for the present, we shall accept that thcre
is a suitablc dual complex and see thc meaning of Maxwell’s second equation
as applied to this dual.
Beginning with space-time notions, we write
∂λ∧F∗
µν =λµνα jα(12)
where jαis the currcnt density. In integral form this becomes
ZC(3)
∂λF∗
µν dxλ∧dxµ∧dxν=ZC(3)
λµνα jαdxλ∧dxµ∧dxν(13)
To visualise this we say that the lines of current correspond to the boundaries
of the F∗
µν varieties. Or, to simplify, let us take a three-dimensional sub-
space. The F∗
µν will now correspond to lines of force and the charges to their
boundaries. Thus charge will come out naturally as discrete. Maxwell’s
second equation thus means that the total charge inside a given region is
equal to the number of lines of force whose boundaries are in that region.
We now return to the consideration of the full four-dimensional space-
time. In terms of chains, Maxwell’s second equation can be written as
(B∗f(2), C(3) ) = (j(2), C(3) ) (14)
Thus the current density corresponds to an abstract I-chain which is the
boundary of a 2-chain.
This completes the expression of the laws of classical electrodynamics in
terms of chains and cochains.
5 Conclusions.
We have introduced a new form of description in which wholeness, activity
and discreteness are given fundamental relevance. Both quantum mechan-
ically and classically, the division between the observing instruments and
14
the observed content no longer arises in our description. The same general
form of language can be used both to describe the experiments and to make
theoretical inferences from them.
Formally, the use of chains and cochains of simplicial complexes is a
natural extension of the informal language from which we started. Using
this form of mathematics we showed how one could describe an experiment,
quantum mechanically or classically, in terms that are not basically different
from those needed in the description of the ‘observed results’. In particular,
as an illustrative example, we put the laws of classical electro-dynamics in
terms of cohomology theory and thus obtained a natural interpretation of
the discreteness of charge. At the same time we showed that the laws were
now in a form that is independent of whether or not there is an underlying
space-time continuum.
In later papers we are going to extend this type of description so as to
include the theory of gravitation and elementary particles. In doing this,
we shall go into more detail as to how the necessary mathematics is to be
developed.
6 References.
Atkinson, D. and Halpern, M. B. (1967). Journal of Mathematical Physics,
6, 373.
Bohm, D. (1960). Quantum Theory. Prentice-Hall, New Jersey.
Bohm, D. (1965a). The Special Theory of Relativity. Benjamin, N.Y.
Bohm, D. (1965b). The Special Theory of Relativity, Appendix. Benjamin,
N.Y.
Bohm, D. (1968). Contemporary Physics, Vol. II, 439 Trieste Symposium
(London).
Bohr, N. (1934). Atomic Theory and the Description of Nature. Cambridge
(reprinted 1961).
Bohr, N. (1958). Atomic Physics and Human Knowledge. Wiley (New
York).
Cairns, S. S. (1961). Introductory Topology. Ronald Press, New York.
Hiley, B. J. (1968). Quantum Theory and Beyond. Cambridge.
15
Hill, E. L. (1955). Physical Review,100, 1780.
Hilton, P. J. and Wylie, S. (1960). Homology Theory. Cambridge.
Misner, C. W. and Wheeler, J. A. (1957). Annals of Physics,2, 525.
Schild, A. (1949). Canadian Journal of Mathematics,1, 29.
Snyder, H. S. (1947). Physical Review,71,38.
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