Karl Svozil
Karl Svozil
Atominstitut der Österreichischen Universitäten
Schüttelstraße 115
A-1020 Vienna, Austria
and
Institute for Theoretical Physics
Technische Universität Wien
Karlsplatz 13/136, A-1040 Vienna, Austria
Tesselation of the quantum field and local connectivity
of the cellular automaton type are investigated.
A connection between this class of models and
lattice gauge theory is established.
Nonlocal models are briefly discussed.
1.
The introduction of cellular automata in theoretical
physics1 initiated several ingenious speculations2-6
concerning their applicability to field theory;
culminating in a sort of ``atomization'' of a continuous
field function j(t,[x\vec])
into discrete lattice states
|
j(t, |
®
x
|
)® j(i)n,[n\vec] (1) |
|
of locally [usually nearest neighbor] connected array elements
evolving in time steps.
This procedure is often called tesselation; i as well
as n and [n\vec] are discrete time and space parameters.
2.
The intention has been not just to
simulate a continuous field phenomenology,
but rather to consider field atomization as factual.
According to this approach,
the perception of a seemingly continuous field is caused by
insufficient experimental space and time resolution.
Hence, given a sufficiently large number
of locally connected array processors
[10100 per cubic meter or so5],
it would be possible to reproduce field theory
exactly by discretizing the states
j(i) as well as the coordinates
n,[n\vec] of the continuous field j(t,[x\vec]).
The amount of information per unit volume of space-time
would be finite.
The macroscopic time evolution would consequently
be the result of some local microscopic cellular automata
rule t symbolized by
|
jn+1,[n\vec] = t(jn,{ [n\vec]} ), (2) |
|
where jn,{ [n\vec]} characterizes a n-dimensional
neighborhood [naively 3-dimensional],
including the kernel jn,[n\vec] itself.
As noted earlier, the field function j takes on
a countable [or even finite] number
N = card({ j(i) } ) of states.
3.
The following observation concerning the topology
of tesselations will be useful:
when the n-dimensional cells [into which space-time
is divided] are contracted to singular points such that their
connections [topologies] are kept, a dual lattice
is produced [see Fig. 1].
From a topological point of view,
the lattice representation is equivalent to a cellular
automaton.
In that way, results from lattice theory can be readily
applied to cellular automata theory and vice versa.
4.
It will be argued, that field theory cannot
be discretized in the way envisaged above.
However, one way to surmount difficulties would
be to give up the condition of locality.
The major support for the argument comes from
a no-go theorem7-9, stating
that for a very general class of fermion lattice
theories under ``mild assumptions'',
it is not possible to construct unambiguous
dispersion relations,
such that for vanishing energy there is only
one allowed value of the momentum.
Usually, additional fermion states would appear,
hampering a lattice model by overloading its
phenomenologic content [``species doubling''].
The assumptions of the no-go theorem are:
a local, homogeneous [translation
invariant on the lattice]
hermitian hamiltonian,
bilinear in the fermion fields; and
locally defined, conserved and quantized
charges bilinear in the fermion fields [such as
the chiral charge].
The dilemma established by the no-go theorem is as follows:
there can be no local, unitary, charge conserving
theory on the lattice without
equal numbers of left- and righthanded fermions for each
combination of quantum numbers [charges].
But
the phenomenology we perceive in nature [for instance in the electroweak
interactions, the hypercharges 2Y(nL) = 2Y(eL) = Y(eR) = 1],
conserves charges, is unitary and has no extra fermions
[such as 2Y(nR) = 2Y(eR) = Y(eL) = 1]
emerging from putting the field on the lattice7.
Hence, lattice models and therefore also cellular automata concepts fail in this context.
5.
One obvious way out of this dilemma is to give up
locality of the transition law (2).
In what follows a promising lattice fermion model is
discussed, satisfying all other conditions
despite locality10-13.
The basic technical ingredient is the application of the finite
element method to quantum field theory.
There, undifferentiated fields appear as averages,
and derivatives as forward differences on the lattice.
As a result, the free Dirac equation
(ig¶-m)y(x) = 0 becomes13
[Planck's constant h = 1 and the velocity of light c = 1,
if not denoted otherwise]:
|
|
ì
í
î
|
i |
3
å
m = 0
|
gm |
å
{ e}
|
|
(-1)em
a(m)
|
- |
m
2
|
|
å
{e}
|
|
ü
ý
þ
|
y(n-e) = 0. (3) |
|
Here, a(m) is the lattice parameter in the m-direction,
em is 0 or 1 for each
m = 0,¼,3,
and (n-e) stands for
(n0-e0,¼,n3-e3), nm Î \rlapI\rlapIN.
With the help of the Fourier transformation
|
y(n) = |
1
(2p)4
|
|
ó
õ
|
p
-p
|
dw |
ó
õ
|
p
-p
|
d |
®
p
|
|
~
y
|
(E, |
®
p
|
)ei(a0n0E-a[n\vec][p\vec]), (4) |
|
where a(j) » const. = a has been assumed,
the following dispersion relation is obtained:
|
tan2( |
Ea0
2
|
) = |
3
å
j = 1
|
( |
a0
a
|
)2tan2( |
pja
2
|
)+( |
ma0
2
|
)2, (5) |
|
which in the limit E, pj << 1 reduces to
Furthermore, it can be shown,
that the equal-time anticommutator
relations are satisfied13:
|
{yf (n0, |
®
n
|
),y(n0, |
®
m
|
)} = |
d[n\vec],[m\vec]
(2p)3
|
. (7). |
|
Given all field values y(n0-1,[n\vec])
at a time n0-1, and the spacial Fourier transform
[(y)\tilde](n0-1,[p\vec]), y(n)
at the next time step n0 is explicitly given by
|
y(n) = |
1
(2p)3
|
|
ó
õ
|
p
-p
|
|
~
D
|
(p) |
~
S
|
(p) e-i[p\vec][n\vec]dp, (8) |
|
where after some calculation13,
|
|
~
D
|
(p) = - |
|
ig0/a0+ |
3
å
j = 1
|
[gj/a]-1tan(pj/2)+1m/2 |
|
|
å
{ e}
|
ei[p\vec][(e)\vec] [a0-2+ |
3
å
j = 1
|
a-2tan2(pj/2)+(m/2)2] |
|
; |
|
|
|
~
D
|
(p) = |
ì
í
î
|
ig0 |
å
{ e}
|
|
ei[p\vec][(e)\vec]
a0
|
-i |
3
å
j = 1
|
gj |
å
{ e}
|
|
(-1)ej
a
|
ei[p\vec][(e)\vec] +1 |
m
2
|
|
å
{ e}
|
ei[p\vec][(e)\vec] |
ü
ý
þ
|
|
~
y
|
(n0-1, |
®
p
|
). |
|
This transition law is nonlocal, since for
the determination of [(y)\tilde],
knowledge about all cells
y(n0-1,[n\vec]) is necessary.
Furthermore, the complex field amplitudes [(y)\tilde]
in (8) are no discrete state functions.
Since locality and discrete states are indispensable
criteria for cellular automata,
above model resembles more an array of [fictive]
real number [not only floating point] processor elements
with infinite storage capacity,
each cell connected to any other cell of the array.
6.
In what follows,
a discretisation of the field amplitude y
is proposed,
such that the energy and momentum variables take on discrete
values per site (n,[n\vec]),
in particular
where k0,kj Î \rlapI\rlapIN and e0,p Î \rlapI R.
Furthermore, a quantum condition on the product
of canonical conjugate parameter pairs (a0,e0) and
(a,p) could be imposed:
where unity stands for Planck's constant.
The discrete field states can the be defined by the state
function y(n0,[n\vec],k0,[k\vec]),
very much resembling the Wigner function.
By defining m: = 2tan-1(m/2),
dispersion relation (5)
for some motion along a pj-axis reads
|
tan2(k0/2) = tan2(kj/2)+tan2(m/2). (11) |
|
For the low-momentum regime kj = 0, k0(kj = 0,m) = m,
whereas for the ``massless'' case
m = 0, k0(kj,m = 0) = kj.
However, a physical interpretation of this procedure
remains to be given.
7.
Another way out of the dilemma established
by the no-go theorem may be the abandonment of
[locally defined] point particles and their associated
quantum numbers [charges etc.].
For extended ``smeared out'' charges5 the no-go
theorem is inapplicable, and fermion doubling could be
avoided14.
8.
A third possibility to circumvent the no-go theorem
may be a higherdimensional configuration space
with dimension D > 4.
In this case, it has to be assumed that for
some reason [yet unknown],
dimensional reduction to D = 4 occurs, such that the
fourdimensional phenomenology is a ``shadow'' of a
higherdimensional world.
I shall refer to this as ``dimensional
shadowing''.
The projection of a higherdimensional local lattice
[with nearest neighbor connections]
onto a lowerdimensional space,
yields nonlocal connections [topologies] in the
latter one, see Fig. 2.
More precisely, when there exists a density
ni(Vi(D)) of point particles per D-volume
element Vi(D),
and a finite overall number of particles
N = åini(Vi(D)) = const.,
then
limD® ¥{ni(Vi)-d(D)i} » 0.
In particular, any finite number N
of points can be rearranged, such that there exists a
``critical dimension'' D* = N-1
associated with a N-simplex,
for which there is exactly one point per D* -dimensional
volume element, and all points are locally connected.
Projection of this D* -dimensional lattice onto any
hyperspace with D < D* ,
yields nonlocal connectivities.
Hence, in order to be able to reproduce connectivity
of every cell with all other cells of the infinite array
of (3)-(8) by dimensional shadowing, D* = ¥.
9.
It has been shown, that putting a local fermion field theory
on a tesselated space equivalent to its dual lattice,
yields well known problems of species doubling.
There may be several approaches to circumvent these obstruction.
Three of those have been discussed:
(i) giving up the locality of the transition law. This has been
discussed in the framework of recent developments in the
theory of finite elements.
The approach would get support from the nonlocality
assumption in the epistomologic interpretation of the
EPR paradoxon.
(ii) giving up local definition of quantum numbers which appear
as ``smeared out'' charge sources etc.
This method is very similar to saying that a particle is a
composit of several cells;
(iii) dimensional shadowing from a highdimensional
[presumedly ¥-dimensional] space onto
fourdimensional configuration space.
The topological equivalence between a tesselated space
and cellular automata on the one hand,
and lattice field theory on the other hand,
allows one to apply theorems established in either area of
research to the other one.
For instance, it immediately follows
that a local lattice field theory is a universal computer,
a result established for continuum field theory recently15.
Moreover, local lattice field theory is able to
construct any embedable field configuration
[in particular replicas], and then set this
configuration free.
Whether such a field can produce
configurations more sophisticated than itself
out of its own and with no specification from the
outside, seems to be an unanswered question
at present.
The author acknowledges discussions with
Anton Zeilinger.
This work was supported by BMWF, project number 19.153/3-26/85.
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.
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.
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\rlapI R2 is tesselated into square cells.
Borders are indicated by bold lines.
After contraction, the twodimensional cubic dual
lattice forms, indicated by dashed lines and bold center points.
Example for dimensional shadowing of \rlapI R2
onto \rlapI R1: whereas in \rlapI R2, A, B, C are locally connected,
their projections p(A) and p(B) are nonlocally connected.
File translated from TEX by TTH, version 1.94.
On 9 Sep 1999, 15:47.