It is a fact that the duality concept is everywhere in both mathematics and physics. Of course, since the list of examples of this fact is very large and since we are concern with quantum gravity let us just briefly mention, as examples in which the duality concept plays a fundamental role, matroid theory [1, 2] [see also [3–9] and references therein] and surreal numbers [10–12] in mathematics and string theory [13] and loop quantum gravity [14] in physics. The origin of matroid theory can be traced back to graph theory were according to the Kuratowski theorem a graph has a dual if does not contain the complete graphs K₅ and K_{3, 3} (see [15]). A matroid is a generalization of the graph concept in which every matroid has a dual. One may understand why matroid theory is a generalization of graph theory by associating with every graph G a matroid M(G). So one must have M(K₅) and M(K_{3, 3}), but according to matroid theory one must have the corresponding duals M*(K5) and M*(K3,3) which turns out to be non-graphic. A surreal number x = {X_L, X_R} is written in terms of the dual sets X_L left set and X_R the right set which satisfies two main axioms (see below). Surprisingly these dual numbers contains the structure of real numbers among other numerical structures. On the other hand it is known that the origin of M-theory [16] was inspired by trying to make sense of a number of dualities between string theory and p-branes. For instance, in eleven dimensions the 1-brane is dual to the 5-brane (see [16]). Finally, it is known that loop quantum gravity emerges from the discovery of the Ashtekar variables which in turn arises by the requirement of the canonical formalism applied to the self-dual Ricci curvature tensor [see [14] and references therein].

Of course, the duality concepts refereed above may be at first sight quite different for each example. So the first step it is to introduce a formal definition of the concept of duality. It turns out that at least in matroid theory one finds such a formal definition [17]. Let M denote the family of all matroids M which corresponding to the ground set E. The matroid duality is a map *:M→M satisfying the two main axioms:

Inspired by this definition of duality in oriented matroid theory let us propose a general tensor definition of duality structure. Consider a family A of all completely antisymmetric tensors A (p-forms), which correspond to space of dimension d, together with an operation + which can be any well-defined tensorial sum operation. The pair (A,+) determines a dual structure through the map *:A→A if satisfies the following axioms:

Note that (II) plays the role of (b) in matroid theory.

Assuming the particular case that A corresponds to family of zero-rank tensors one may add two additional axioms, namely

One can prove that the element 0 in (III) is unique as follows: Assume that (A,+) is a dual structure with two self-dual elements 0 and 0′. Then 0 = 0+0′ = 0′. Moreover, according to the axiom (IV) the element ^*A can be considered as the inverse of A. In order to show that the inverse ^*A is unique one takes recourse of the axiom (I) instead of the associativity axiom in group theory. In fact, assume that an arbitrary element A in A has two inverses ^*A and ^*B. Thus, one has (i) A+ ^*A = 0 and (ii) A+ ^*B = 0. Applying the axioms (I) and (III) to (ii) one obtains ^*A+ B = 0 and therefore according (i) one gets ^*A+ B = ^*A+A which means that B = A. The two axioms (III) and (IV) are similar to the definition of a field in number theory. For these reasons one it is straightforward to verify that the integer Z and the real numbers R are in fact dual structures.

The main goal of the present work is to comment about the possibility that the various concepts such as oriented matroids, qubits, twistors, and surreal numbers are linked by the duality symmetry. Moreover we shall argue that such a dual concept may be considered as an underlying mathematical tool in quantum gravity.

It turns out that the completely antisymmetric ε-symbol becomes the underlying mathematical object in all these connections. Specifically, the ε-symbol can be defined as

Here, the indices a₁, …, a_d run from 1 to d. This is a d-rank density tensor which values are +1 or −1 depending on even or odd permutations of ε^12…d, respectively. Moreover, εa1…ad takes the value 0 unless the values of a₁…a_d are all different. Lowering and rising the indices with a Kronecker delta δ_ab one finds that

where δb1…bda1…ad is a generalized Kronecker delta. A contraction in (2) of the last n-indices of the type a_i with the last n-indices of the type b_i leads to

with n = d−k+1. In particular one has

Let vai be any d × n matrix over some field F, where the index i takes values in the ordinal set E = {1, …, n}. Consider the object

Using the ε-symbol property

it is not difficult to prove that Σi1…id satisfies the Grassmann-Plücker relations [see [18] and references therein], namely

Here, the brackets in the indices of (6) and (7) mean completely antisymmetric.

Through (5) one can define the object

where e_i1, e_i2, …, e_id are 1-form bases associated with the (dn)-dimensional real vector space of alternating d-forms on Rⁿ. It turns out that (8) can also be written as

for some v1,v2,…,.vd∈Rn. This means that Σi1…id corresponds to an alternating decomposable d-form [19].

A realizable chirotope χ is defined as

In order to define non-realizable chirotopes it is convenient to write the expression (7) in the alternative form

where

Here, j_d+1 = i_d and ȷ^k establish the notation for omitting this index. Thus, for a general definition, one defines a d-rank chirotope χ:E^d → {−1, 0, 1} if there exist r1,…,rd+1∈R+ such that

with

and k = 1, …, d+1. It is evident that (11) is a particular case of (13). Therefore, there are chirotopes that may be non-realizable. Moreover, this definition of a chirotope admits a straightforward generalization to the complex structure setting. In this case the complex chirotopes are called phirotopes [20–22].

Given a chirotope (or phirotope) χi1…id its dual is defined as

Here D = d+p is the total dimension of the ground state E. Observe that due to the relations (3) one gets

which means that χ satisfies the axiom (I). It turns out that (16) is true for a general completely antisymmetric object F (d-form) when its dual is defined in terms of the ε-symbol. In fact, when D is even one can write D = d+d = 2d and in this case one can define the self-dual (antiself-dual) tensor as

One observe that ^±F satisfies

Thus, one sees that for D even the ^±F tensor is another self-dual (antiself-dual) notion other than the 0 element in the axioms (III) and (IV).

Let us now explain how the Grassmann-Plücker relation (7) is connected with qubit theory [see [23] and references therein]. For this purpose consider the general complex state ∣ψ>∈C^2N

where the states ∣A₁A₂…A_N> = ∣A₁>⊗∣A₂>…⊗∣A_N> correspond to a standard basis of the N-qubit. For a 3-qubit (19) becomes

while for 4-qubit one has

It turns out that, in a particular subclass of N-qubit entanglement, the Hilbert space can be broken into the form C^2N = C^L⊗C^l, with L = 2^N−n and l = 2ⁿ. Such a partition allows a geometric interpretation in terms of the complex Grassmannian variety Gr(L, l) of l-planes in C^L via the Plücker embedding. It turns out that in this scenario the complex 3-qubit, 4-qubit admit a geometric interpretation in terms of the complex Grassmannian varieties Gr(4, 2), Gr(8, 2), respectively [see [23] for details]. The idea is to associate the first N−n and the last n indices of Q_A1A2…AN with a L×l matrix ωa1i1. This can be interpreted as the coordinates of the Grassmannian Gr(L, l) of l-planes in C^L. Using the matrix ωp1i1 one can define the Plücker coordinates

which one recognizes as the complex version of the decomposable tensor Σi1…id defined in (5). Moreover, one verifies that under the transformation ω → Sω with S∈GL(l, C) the Plücker coordinates transform as Qi1…id→Det(S)Qi1…id and of course ±i1…id satisfies the Grassmann-Plücker relations

Now, consider the quantity σμ=(σ0,σ^), where the σ^ denotes Pauli matrices and σ₀ is the identity matrix. By using σ_μ the linear momentum in 4-dimensions p^μ can be written as

This is the spinorial representation of p^μ. An interesting aspect emerges if one sets Det(p^AḂ) = 0, corresponding to a null momentum pμpμ=0. This means that

A solution to this equation is given by p^AḂ = ξ^Aη^Ḃ. Since p^μ is real vector one verifies that pAḂ=p̄ḂA and therefore

One finds that this last expression means that ηḂ=aξ̄Ḃ, where due to (26) one sees that a is real number which can be normalized in the form a = ±. So one has found that, in the case of null momentum, one can write p^AḂ in terms of a more fundamental complex quantity ξ^A, namely

Similar analysis applies to the total angular momentum M^μν = x^μp^ν−x^νp^μ+S^μν, where S^μν is the internal angular momentum satisfying the Tulczyjew second class constraint [24];

Observe that due to (28) and since p^μ is a null vector one has Mμνpν=−(xνpν)pμ This means that δαβγτμνpαMβγpν=0. In turn this leads to εσαβγεστμνpαMβγpν=0 or εσαβγεστμνpαSβγpν=0. Therefore, if one defines the 4-vector Sσ=12εσαβγpαSβγ one obtains εστμνSσpν=0 and consequently one discovers that

for some non-vanishing constant s which is identified with the helicity of the system. This means that the spin S_μ is parallel or anti-parallel to p_μ depending of the sign of s. So, determining p^AḂ in terms of ξ^A via (27) is equivalent to determine S^AḂ in the form SAḂ=sξAξ̄Ḃ. Thus, considering (28) one sees that the left relevant part of M^μν is

but again since p^μ is a null vector one has Lμνpν=−(xνpν)pμ which means that out of the six true degrees of freedom of L^μν = −L^νμ three are already determined by p^μ. Therefore, the corresponding spinor representation of L^μν, namely LAḂCḊ=σμAḂσνCḊLμν, can be written as

Here, μ^AC = μ^CA is a symmetric matrix and therefore has only three independent complex degrees of freedom. In order to reduce μ^AC to only three real components which of course are related to the true three degrees of freedom of L^μν one further writes μ^AC in the form μ^AC = ξ^Aπ^C+ξ^Cπ^A. If to the coordinates ξ_Ȧ one adds the the spinor π^A one is lead to the twistor structure Pα=(πA,ξȦ) [25] [see [26] and references therein] which can be identified with a point in C⁴. This analysis revel that in the case of a null system the eight coordinates (x^μ, p^ν) in R⁸ may in principle be associated with the coordinates (πA,ξȦ) in the twistor complex space C⁴.

Consider the 2-index twistor

which can also be written as

If one defines p1μ=xμ and p2μ=pμ one sees that L^μν can be written as

and therefore one concludes that Pαβ can be understood as the complexification of L^μν. Of course, Pαβ satisfies the Grassmann-Plücker relations

It turns out that Pαβ can be used to associate points in C⁴ with points in the complexified Minkowski space (see [25]). From the perspective of oriented complex matroids, Pαβ is just a representable phirotope. One is tempted to assume that a generalization of twistor theory may be also be associated with the phirotope theory.

Is it possible that twistors or qubits can be related to surreal number theory [10, 11, 12]? Consider the set

and call X_L and X_R the left and right sets of x, respectively. Conway develops the surreal numbers structure S from two axioms:

Axiom 1. Every surreal number corresponds to two sets X_L and X_R of previously created numbers, such that no member of the left set x_L∈X_L is greater or equal to any member x_R of the right set X_R.

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 x_L ∈ X_L and x_R ∈ X_R one has x_L  x_R. This is denoted by X_L  X_R.

Axiom 2. One number x = {X_L ∣ X_R} is less than or equal to another number y = {Y_L ∣ Y_R} if and only the two conditions X_L  y and xY_R are satisfied.

This can be simplified by saying that x ≤ y if and only if X_Ly and xY_R.

Observe that Conway definition relies in an inductive method; before a surreal number x is introduced one needs to know the two sets X_L and X_R of surreal numbers. Using Conway algorithm one finds that at the j-day one obtains 2^j+1−1 numbers all of which are of form

where m is an integer and n is a natural number, n > 0. Of course, the numbers (37) are dyadic rationals which are dense in the reals R.

The sum and product of surreal numbers are defined as

and

respectively. The importance of (38) and (39) is that allow us to prove that the surreal number structure is algebraically a closed field. Moreover, through (38) and (39) it is also possible to show that the real numbers R are contained in the surreals S [see [10–12] for details]. Of course, in some sense the prove relies on the fact that the dyadic numbers (37) are dense in the reals R.

In 1986, Gonshor [12] 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 f(1) = +, f(2) = +, f(3) = −, f(4) = + then f is the surreal number (++−+). In the Gonshor approach one obtains the sequence: 1-day

in the 2-day

and 3-day

respectively. Moreover, in Gonshor approach one finds the different numbers through the formula

where ε₀, ε₁, ε₂, …, ε_q∈{+, −} and ε₀≠ε₁. Furthermore, one has ∣+∣ = + and ∣−∣ = −. As in the case of Conway definition, through (43) one gets the dyadic rationals. Just for clarity, let us consider the additional example:

By defining the order x < y if x(α) < y(α), where α is the first place where x and y differ and the convention − < 0 < +, it is possible to show that the Conway and Gonshor definitions of surreal numbers are equivalent [see [12] for details].

Suppose that instead of qubits we consider a rebit (real bits) which can be thought as j-tensor [4],

where the indices A₁, A₂, …, A_j run from 0 to 1. Of course j indicates the rank of t_A1A2…Aj. In tensorial analysis, (45) is a familiar object. One arrives to a link with surreal numbers by making the indices identification 0 → + and 1 → −. For instance, the tensor t₀₀₁₀ in the Gonshor notation becomes

In terms of t_A1A2…Aj, the expressions (40), (41) and (42) read

in the 2-day

and 3-day

respectively.

Note that there is a duality symmetry between positive and negative labels in surreal numbers. In fact, one can prove that this is general for any j-day. This could be anticipated because according to Conway definition a surreal number can be written in terms of the dual pair left and right sets X_L and X_R. Further, the concept of duality it is even clearer in the Gonshor definition of surreal numbers since in such a case one has a functions f with the image in the dual set {+, −}. In terms of the tensor t_A1A2…Ap such a duality can be written in the form

where

The identification of surreal numbers with rebits means that its complexification must be related to qubit theory and therefore with twistor theory. So one has discovered that the use of the completely antisymmetric object epsilon εa1…ad allows to define the Plucker coordinates which must to satisfy the Grassmann-Plücker relation. In turn, we have proved that this relation is a common mathematical central notion in oriented matroids, qubit theoy, twistor theory and surreal number theory.

Moreover, it has been proved in Mosseri and Dandoloff [27], Mosseri [28], and Bernevig and Chen [29] that for normalized qubits the complex 1-qubit, 2-qubit, and 3-qubit are deeply related to division algebras via the Hopf maps, S3→S1S2, S7→S3S4, and S15→S7S8, respectively. It seems that there does not exist a Hopf map for higher N-qubit states. So, from the perspective of Hopf maps, and therefore of division algebras, one arrives to the conclusion that 1-qubit, 2-qubit, and 3-qubit are more special than higher dimensional qubits [see [27–29] for details]. Again one wonders whether surreal numbers can contribute in this qubits theory framework.

The original idea of Penrose was to replace the continuity of the Minkowski space for new geometric framework which may allow for a discrete structure and in this way unify general relativity and quantum mechanics. In fact, one of the original motivation to introduce twistors was be able to have mathematical arena in which the discretization of the spacetime was possible. The hope was that the complex structure of twistors may be connected with quantum mechanics. In a sense the idea was to replace R⁴ by C⁴ and in this way, since the object in C⁴ are complex, one may be able to connect with quantum mechanics which intrinsically is a complex structure. Ironically, according to the discussion in this work, it seems to us that the combinatorial structure searched by Penrose in connection with quantum gravity is not the twistors itself but the underlying oriented matroid theory. But ground set in oriented matroids can be constructed by strings of the set {+−} which are the main tool in qubit theory and surreal numbers. All these comments suggested that the concepts such as chirotopes (phirotopes), qubits, twistors, and surreals must be considered mathematical tools underlying quantum gravity.

Let us analysis deeply the connection between surreal numbers and qubits. For this purpose we shall assume that one may be able to write a surreal complex numbers Z in the form

where J1 and J2 are two surreal numbers according to the formula (43). This complexification of surreal numbers must establish a complete connection with the N-qubit structure if one assume the existence of a complex operator ẐA1A2…AN such that

This is inspired in the observation that J in (43) can be associated with the eigenvalues of a z-component Ĵ_z of the total angular momentum Ĵ in quantum mechanics. Of course in such case one has Jz=l±12, with the identification of 12-spin of the system. The surprise with surreal numbers is that predicts that besides 12-spin system there must exist infinite number of J-spins, according to the formula (43). Thus, for instance one must include particles with 14-spin (see [30, 31]) and 18-spin and in general particles with dyadic rational m2n-spin.

Traditionally, quantum mechanics enter in the above twistor formalism when one writes all possible gauge fields (and their associated field equations) in twistor language and proceed to quantize in the usual way. In the case of qubit theory things are different because, even from the begining, qubits refers to quantum states. Thus, concepts of quantum mechanics such as the density of states are constructed from the corresponding entanglement monotones [23]. Here, we would like to propose an alternative possible route to connect further our formalism with quantum mechanics. The central idea is to continue looking the surreal numbers as a quantities associated with different dyadic spins (m2n-spin). Let us explain in some detail this idea. As we mentioned J in (43) seems to play the analog of the eigenvalues of the z-component Ĵ_z of the angular momentum operator, namely Jz=l±12. Roughly speaking, from the point of view of number theory, the quantization of a physical system means to go from the real numbers (continuum) R to natural numbers N (discrete). In the case of surreal numbers things are different because one starts with the 0-day, 1-day, 2-day, and so on and in the ω-day (this is the way mathematitians called) one obtains the real numbers R. In other words one starts with a discrete structure and finds the continuum scenario. Moreover, if in addition to (43) one uses the identity

it is not difficult to show that J in (43) satisfies the expression

Since l<j one also has

Here, one assumes that from (43) one has j = l+s. Of course, (56) is the analoguos inequality of the eigenvalue of the total angular momentum. Following this route of thoughts one first note that surreal numbers of the type (++…++) (or the corresponding negative part) can be associated with higher integer-spins, 1, 2, 3, …, while surreal numbers of the type (++…+−) can be associated with half-inter spins, 1/2, 3/2, 5/2, …. This means that in principle bosons and fermions are part of the surreal structure and therefore supersymmetry must be present. Thus one must expect that a generalized supersymmetry can be obtained if one includes other surreal numbers such as 1/4, 3/4, 1/8, 3/8, and so on. Since, as we mentioned, the dyadic rational m/2ⁿ are dense in the reals R one should expect that eventually, in the ω-day, the anyons may emerge. What about the graviton? This corresponds to the surreal number 2 or 2-spin. Thus, just as in string theory the graviton is just one resonance out of many or even infinity resonances, in our case the graviton is just a physical system with particular value 2-spin, but in principle one has all kind of dyadic-spin particles. Thus, according to these observations it seems that quantum gravity should not be seen as an isolated problem but as part of a much larger system in which all types of dyadic-spins are present.

Another source of interesting developments it may emerge from the analysis of singularities, both in balck-holes and cosmology. In fact, from the point of view of surreal numbers theory the black-hole singularity 2MG/c²r → ∞, when r → 0, and the Big-Bang singularity (of the radiation energy density) ρr=ρ0/a4→∞, when a → 0 are not a real problem because in such a mathematical theory all kind of infinite large and infinite small are present.

It is worth mentioning that in Atiyah [32], the twistor space and the Plücker coordinates are used to determine the geometry of the instantons solutions of Yang-Mills theory. It may interesting for further research to find the connection between instantons formalism and surreal number theory.

Finally, let us just mention that using fiber bundle concept in oriented matroid theory [33, 34] a connection with p-branes and phirotopes was established [6]. Thus according to the present development one may expect that eventually a link between p-branes and surreal numbers must be route to follow in the quest of quantum gravity.