1. Introduction
In mathematics, from number theory history [1], one learns that, roughly speaking, the starting point was the natural number N and after centuries of thought evolution one ends up with the real number R from which one constructs the differential and integral calculus. In physics, ironically, one starts with classical mechanics which is based on continuous function over the real R and ends up with quantum mechanics whose structure leads to change in our physical continuous observable quantities for discrete values based on the natural numbers N. The main lesson learned from these histories is that the underlying mathematical structure is, indeed, number theory.
The orbital angular momentum L provides us the clearer example of the importance of the underlying number theory. In fact, in classical mechanics, L is considered in terms of the position and the momentum, both of which are assumed to be continuous functions of time over the real R, while in quantum mechanics, in the Bohr model of the hydrogen atom, one has
with
the Planck fundamental constant and
. (In order to give emphasis to the numerical aspect let us just set units such that
.) So, let’s say, one uses L over R in planetary systems, but L over N in atomic scale systems. In any case, one observes that number theory is the basic mathematical notion, somehow hidden in this analysis.
When one makes a reflection on the above comments at the level of fundamental particles one discovers that one needs to add the spin of particles which is in some cases, as the photon, a number in N, while in other cases, such as the electron (
) or the gravitino (
), is a number of the type
with m odd. It turns out that particles with integer spin are called bosons, while particles with half integer spin are called fermions. It is worth mentioning that supersymmetry combines both types of particles: bosons and fermions. The interesting thing is that such numbers, integers and half integers are subsets of the rationals Q which in turn are contained in R. Thus, at the level of fundamental particles one seems to go backward from N to Q and perhaps later on to R, as we shall see below. So, one may ask the question: is it possible to have a mathematical structure that describes this numerical evolution of the spin concept at the level of fundamental particles? Here, we would like to propose the answer: surreal numbers theory.
2. A Short Comment about Surreal Number Theory and Dyadic Rational Spin
Let us briefly describe what a surreal number means. For this purpose consider the set
(1)
and call
and
the left and right sets of x, respectively. In 1973, Conway [2] (see also Ref. [3]) developed the surreal numbers structure
from two axioms:
Axiom 1. Every surreal number corresponds to two sets
and
of previously created numbers, such that no member of the left set
is greater or equal to any member
of the right set
.
Let us denote by the symbol
the notion of no greater or equal to. So the axiom establishes that if x is a surreal number then for each
and
one has
. This is denoted by
.
Axiom 2. One number
is less than or equal to another number
if and only if the two conditions
and
are satisfied.
This can be simplified by saying that
if and only if
and
.
Observe that Conway definition relies in an inductive method; before a surreal number x is introduced one needs to know the two sets
and
of surreal numbers. Using Conway algorithm one finds that at the
-day one obtains
numbers, all of which are of form
(2)
where m is an integer and n is a natural number,
. Of course, the numbers (2) are dyadic rationals which are dense not only in the rationals Q but also in the real R. It is also possible to show that the real numbers R are contained in the surreals
(see Ref. [2] [3] for details). Of course, in some sense, the proof relies on the fact that the dyadic numbers (2) are dense in the real R.
In 1986, Gonshor [4] introduced a different but equivalent definition of surreal numbers:
Definition 1. A surreal number is a function f from initial segment of the ordinals into the set
.
For instance, if f is the function so that
,
,
,
then f is the surreal number
. In the Gonshor approach, one obtains the sequence: 1-day
(3)
in the 2-day
(4)
and 3-day
(5)
respectively. Moreover, in Gonshor approach one finds the different numbers through the formula [4]
(6)
where
and
and
. As in the case of Conway definition, through (6) one gets the dyadic rationals. Just for clarity, let us consider the example:
(7)
By defining the order
if
, where
is the first place where x and y differ and the convention
, it is possible to show that the Conway and Gonshor definitions of surreal numbers are equivalent (see Ref. [4] for details).
Let us focus in (6) with
(and therefore with
). Thus, (6) becomes
(8)
It remains to clarify what index p, in the sum symbol, means. In order to clarify this aspect instead of (8) one may write the more complete expression [5]:
(9)
Notice that according to this expression one always has
.
Similar analysis for
and
in (6) lead us to
(10)
Observe that
. The first thing that one notes is that (I) in (9) and (10) gives the integer numbers
and
. By completeness one set
. While (II) provides with the dyadic rationals
where m is an odd element in the rationals
and
. It turns out that from the point of view of the spin concept, (I) gives integer spin numbers Z corresponding to bosons, while (II) establishes the half integer spin numbers corresponding to fermions. So, one concludes that (I) and (II) in (9) and (10) determine the spin numerical values for bosons and fermions, just the kind of particles considered in supersymmetry. So our contribution to this spin framework must come from (III) in (9) and (10). In order to have a better understanding of what exactly one means by this, we shall consider several examples.
Assume
. In this case (9) becomes
(11)
This implies that from (I) and (II) one gets
,
and from (III) one obtains
,
and so on. Since
one also has
,
and
,
and so on. Hence, according to these results in addition to particles with 1-spin and
-spin obtained from (I) and (II) our prediction is that there must exist particles with
-spin and
-spin and so on. It is interesting that particles with
-spin have already considered in the literature [6] [7] and [8]. But our prediction here is that
-spin gauge fields must be just part of a larger list of new dyadic
-spin particles, with m odd.
An interesting combination between (11) and (9) (and (10)) is
(12)
with
[5]. This means that the tree
(and
) plays the role of a main building block; any other tree
with
can be obtained from (12). Surprisingly,
has been studied in the context of Zeno algorithm [9] and Minkowskis question mark function [10]. In some sense if
were added to
and (12) was used the surreal numbers could be discovered for another routes, different than game theory.
Another interesting aspect of the tree structure
,
and
is that one can derive in an alternative way how many numbers are created in the
-day. It is worth to mention that this notion of “day” is used by the mathematicians in spire of their development of surreal numbers theory is considered only in the mathematical context. First, let us use Gonshor formalism to answer this question. In the 0-day one starts with the number 0 and in the 1-day the numbers −1 and +1 are created, namely (−) and (+). While in the 2-day 4 numbers are created, namely
,
,
,
, and so on. So one has that the series
determines the total numbers of surreal numbers that at the
-day are created. But using the identity
one discovers that
. Now,
is two parameter function
and
. If one sets
and change
one moves vertically producing the corresponding tree, as
. While if one sets
and change
one is moving horizontally. In this sense
determines the day parameter used by the mathematician. As an example set
. From (9) one obtains
,
,
, and the corresponding negatives. So, in the 3-day we have 8 numbers and so one discovers the series
which is what one obtains with Gonshor approach.
3. Surreal Numbers and Rational Dyadic Spin-Field
Until now we have established a link between surreal number theory and the physical concept of spin. A natural question is; how such a numerical scenario can be associated with gauge fields? Consider a plane wave gauge field
. One says that
is a gauge field of spin s if satisfies the transformation rule
(13)
In the case of the electromagnetism it is shown that for a wave propagating in the z-direction the 4-vector potential
, in the Lorentz gauge, leads to the transformation
(14)
with
. While in the case of gravitational gauge field
, again propagating in the z-direction and in the Lorentz gauge, transforms as
(15)
with
. Thus, (14) and (15) are used to say that electromagnetism and gravitation are gauge fields of 1-spin and 2-spin, respectively. From this rough analysis, one sees a patter; if there is just one spacetime index in the gauge field
the corresponding spin is 1, while if there are two indices in the symmetric gauge field
the spin is 2, whereby convenience we define
. Therefore, if one considers a completely symmetric gauge field
it is expected that the associated spin must be
. Just the integer numbers associated with
and the neutral element
predicted in the surreal numbers (I) given in (9) and (10).
We would like now to consider the case of
-spin gauge spinor field. In this case one has a complex spinor
which of course satisfies de Dirac equation [11]
(16)
Here, as it is known, the
-matrices satisfy the Clifford algebra
(17)
with
and
.
Comparing
with
one notes two important differences;
is complex and the index a is spinorial, while
is real and contains the target space-time index
. The complexity of
can be solved by imposing a Majorana condition
, where
is the charge conjugation matrix. Similar observation applies to a field of the form
which only contains the spacetime index
. So according to the previous discussion although both fields
and
are continuous function over the real their associated spin structure is discrete. What if one now combines indices in the form
? For
one imposes, of course, the Majorana condition. The question emerges what is it the spin associated with the gauge field
? It is not difficult to guess that
is a field of
-spin (gravitino field). If we now consider a Majorana gauge field of the form
one must find that this corresponds to a
-spin field. But one has
, just as the surreal numbers in (II) in (9) and (10). Summarizing the
-spin (integer spin) structure of
corresponds to (I) in (9) and (10), and
with
-spin (half integer spin) structure corresponds to (II) in (9) and (10). So, one can say that (I) and (II) correspond what in the literature is called “higher spin theories” (see Ref. [12] and references therein). Again the question emerges what could be the gauge field associated with (III) in (9) and (10)?
In order to answer the above question let us again observe the indices in the gauge fields
and
. One notes that, only because one introduces an appropriate spinor index a, the numerical nature of the spin changes drastically from integer
to half integer
. Observe also that the number
enter (in the two first numbers) also in (III). So one should expect that in order to include lets say terms like
-spin one must introduce a new type of index in the gauge field
. In fact, one finds that a gauge field of the form
makes the job. But immediately the question arises: What could be the field equation that
must satisfy? One can explore first
which we simply write as
. In the case of
the field equation is, of course, the Dirac Equation (16). It turns out that this equation is so important in quantum mechanics and supersymmetry that it seems difficult to modify in order to make some room for the field
. Fortunately, we have a possible proposal and this goes with the name of Elko field equation [13] [14].
First, recall that Dirac equation (16) can be considered as the “square root” of the Klein-Gordon equation. Surprisingly, in 2005 Ahluwalia and Grumiller [13] [14] proved that such a “square root” is not unique. In fact, assuming more general helicity eigenvalues they proved that an alternative and different field equation
(18)
emerges which can also be reduced to the Klein-Gordon equation. They call their formalism Elko theory [13] [14]. Here, the indices
, etc. run from 0 to 3, the indices
run from 1 to 2 and the spinorial indices
run from 1 to 4, with
is the totally antisymmetric
-symbol, with
. One of the interesting aspects of the Elko fields is that they provide one of the most interesting candidates for dark matter [15].
In contrast to the usual Dirac equation, (18) requires 8-component complex spinor field
rather than 4-component
which is the case in equation (3). This is solved by imposing in
the kind of Majorana condition. Furthermore, the quantities
and
in (1) establish that
is not eigen-spinor of the
operator as
in the Dirac equation (16).
It turns out that (18) can be generalized to the form [16]
(19)
where
is the generalized Kronecker delta which is related to
by
(20)
It seems that
-rank totally antisymmetric tensor spinor fields of the form
, is what one is looking for. An interesting feature of our formalism arises when one requires that the field
satisfies the Grassmann-Plücker relations
(21)
for each value of the index a in
. Here, the bracket
means totally antisymmetric. Thus, one finds that due to (21) the field
is decomposable and therefore can be written as
(22)
where the only non-vanishing terms of
are
. The Majorana condition for
looks like
(23)
One recognizes in (22) a kind of generalization of the Plücker coordinates for
. Hence, (22) determines one-to-one correspondence between the fields
and
. Using (23) one can make identification
. Thus, it seems that
is the gauge field that one is looking for in order to make sense of
-spin gauge field, but further work is required.
Just for completeness consider a complex state
expressed as (see Ref. [17] and references therein)
(24)
This is a N-qubit sate, where
correspond to a standard basis. In a particular subclass of N-qubit entanglement, the Hilbert space can be broken into the form
, with
and
. Such a partition allows a geometric interpretation in terms of the complex Grassmannian variety
of
plains in
via the Plücker embedding. The idea is to associate the first
and the last n indices of
with a
matrix
. This can be interpreted as the coordinates of the Grassmannian
of
-plains in
. Using the matrix
one can define the Plücker coordinates
(25)
which one can associate with (25) by making the identification
. Just as in qubit theory the transformation
, with
, the Plücker coordinates transform as
(see Ref. [17] for details) one discovers that under
the field
transform as
. This shows that according to our previous discussion that the Plücker coordinates may be the key concept for surreal numbers framework and therefore in
-spin gauge field theory. It turns out that the Plücker coordinates are also a key concept in one of the possible definitions of realizable oriented matroids [18] (see also Refs. [19] - [26] and references therein).
Surprisingly, the rebits (the real part of the complex qubits) can be connected surreal numbers and therefore one can also associate the Plücker coordinates of
with the surreal numbers. As we mentioned using Conway algorithm one finds that at the “
-day” one obtains
numbers all of which are of form
, where m and n are integers numbers. Of course, these numbers are dyadic rationals which we already know that are dense in the real R. The mathematicians call
-day when the surreal numbers lead to the real R. It is not difficult to prove that in the
-day the tree
leads to the interval
over the real R. In this context, one finds that in the
-day one must associate a continuous spin over the real R. These types of continuous spins are of course very well known; particles with such spin are called anyons. The surprisingly thing is that the creation of surreal numbers do not finish in the
-day, but can continue let’s say to the
-day,
-day, etc. What this means in the framework of physical spin is at present mysterious for us.
4. Surreal sl(2,r) Correspondence
It is worth mentioning that in Ref. [22] a connection between oriented matroid theory and super p-branes was established. Thus, according to the present development, one may expect that eventually a link between super p-branes and surreal numbers may appear. Therefore one is tempting to believe that our proposed surreal spin theory may emerge as a key structure in the quest for quantum gravity. A strongly evidence for this conjecture may come from the observation that
-spin and
-spin particles (semionic) have already been considered in a kind supergravity theories [7]. The relevant symmetry, in this case, is the
group which has already been connected with superstrings [27] [28]. Let us express from the point of view of the surreal number structure the relevance of this group in superstrings.
Let us first introduce the generators
of the group
which is isomorphic to
,
or
(26)
which are of course related to the Pauli matrices in the form
and
. Thus, we find that the
satisfies the algebra
(27)
and also
with
. As usual, in order to discuss any representation of
one introduces the raising and lowering operators
(28)
Thus, one finds that the Casimir operator
(29)
becomes
(30)
With these algebraic tools, the unitary irreducible representation of
can be obtained. Typically one has discrete and continuos representations, depending of the eigenvalues k of
and the eigenvalues q of the Casimir operator
. Here, however we are more interested in considering the exceptional representation
(31)
where one assumes that
and a satisfies the typical expressions of the harmonic oscillator, namely
(32)
with
. Observe that
(33)
It is not difficult to see that (31) is consistent with (27). From (30), (31) and (32) one obtains
(34)
and
(35)
and therefore we discover that
(36)
This means that the combination
(37)
leads to the formula
(38)
with
(39)
In the minimum case, the solution of (38) corresponds to
and
. Since
(40)
annihilate the vacuum for
and
, that is
and
the whole solution for
can be separated in the even part
(41)
corresponding to
and odd part
(42)
for the case
. One can make a contact with surreal numbers if one now observes that
,
, obtained from (III) in (9) and (10). But from (12) one gets
(43)
This means that
(44)
So, by taking
(45)
one obtains the two eigenvalues (41) and (42) corresponding to
and
.
5. Surreal Superstrings?
Now we would like to pursue the idea of using the previous surreal
correspondence in order to develop a kind of surreal superstring theory. Fortunately, there are a number of works, including “Cosets as gauge slices in
strings” by Hwang [27] and “Strings in AdS3 and the SL(2, R) WZW Model; 1, 2, 3” by J. Maldacena and H. Ooguri [28] among others, that establishes a bridge between the group
and strings. The starting point in these developments is to Fourier expand the generators of
in the form
(46)
with
and
(47)
We require that the generators
satisfies the algebra
(48)
(49)
(50)
and the same for
. Of course, this algebra corresponds to the Hilbert space of the WZW model. It is not difficult to see that the origin of this algebra can be traced back to the algebra of the generators of
, namely the expressions (27). For this reason this algebra we shall called
.
The Virasoro generators are defined by
(51)
and
(52)
which presumably satisfies the Virasoro algebra
(53)
where
(54)
is the central charge.
6. Final Remarks
From the above development, it becomes evident that the Fourier series of the generators (31) leads to straightforward contact with string theory. However, the eigenvalues
and
which are predicted by (31) become hidden spin structures. This is part due that our discussion has been based on a bosonic perspective but in fact from the original discussion one knows that semionic particles with spin
and
behaves in very peculiar way. In particular, Zee [6] has studied the idea that the quasi-particles in the superconductors may be semions. Similarly, Sorokin and Volkov [8], using twistor-like formulation, obtained supersymmetric equations for semions with spin
and
. Finally, Mezincescu and Townsend [7] develop the theory of semionic supersymmetric solitons. These works show, among other things, that the particles with spin
and
may be relevant at the supersymmetric level. As far as we know, besides the article [26], until now there seems not to be works considering
-spin or in general dyadic
-spin and therefore it turns out that surreal supersymmetry remains as an open problem. But at least from the above comments it becomes clear that surreal supersymmetry and surreal p-branes must be the relevant route for further work.
Moreover, the present work may be of special relevant interest to be connected with other related subjects in theoretical physics scenario such as the ones described in the Refs [29] and [30] in which it investigated the generation of 1-spin field and the Pythagoreans figurative numbers, respectively.
Acknowledgments
JA Nieto would like to thank the Mathematical, Computational & Modeling Sciences Center of the Arizona State University where part of this work was developed. This work was partially supported by PROFAPI-UAS.