Quantized Fields\A La Clifford and Unification

$Quantized Fields\A La Clifford and Unification$ Quantized Fields àla Clifford and Unification1 Matej Pavˇsiˇc JoˇzefStefan Institute, Jamova 39, 1000 Ljubljana, Slovenia e-mail: [email protected] It is shown that the generators of Clifford algebras behave as creation and annihilation operators for fermions and bosons. They can create extended objects, such as strings and branes, and can induce curved metric of our spacetime. At a fixed point, we consider the Clifford algebra Cl(8) of the 8-dimensional phase space, and show that one quarter of the basis elements of Cl(8) can represent all known particles of the first generation of the Standard model, whereas the other three quarters are invisible to us and can thus correspond to dark matter. 1 Introduction Quantization of a classical theory is a procedure that appears somewhat enigmatic. It is not a derivation in a mathematical sense. It is a recipe of how to replace, e.g., the classical phase space variables, satisfying the Poisson bracket relations, with the operators satisfying the corresponding commutation relations [1]. What is a deeper meaning for replacement is usually not explained, only that it works. A quantized theory so obtained does work and successfully describes the experimental observations of quantum phenomena. On the other hand, there exists a very useful tool for description of geometry of a space of arbitrary dimension and signature [2]{[7]. This is Clifford algebra. Its generators are the elements that satisfy the well-known relations, namely that the anticommutators of two generators are proportional to the components of a symmetric metric tensor. The space spanned by those generators is a vector space. It can correspond to a physical space, for instance to our usual three dimensional space, or to the four dimensional spacetime. The generators of a Clifford algebra are thus basis vectors of a physical space. We will interpret this as a space of all possible positions arXiv:1707.05695v1 [physics.gen-ph] 17 Jul 2017 that the center of mass of a physical object can posses. A physical object has an extension that can be described by an effective oriented area, volume, etc. While the center of mass position is described by a vector, the oriented area is described by a bivector, the oriented volume by a trivector, etc. In general, an extended object is described [8, 9, 10, 11, 12] by a superposition of scalars, vectors, bivectors, trivectors, 1A chapter in the book Beyond Peaceful Coexistence; The Emergence of Space, Time and Quan- tum (Edited by: Ignazio Licata, Foreword: G. 't Hooft, World Scientific, 2016) 1 etc., i.e., by an element of the Clifford algebra. The Clifford algebra associated with an extended object is a space, called Clifford space2. Besides the Clifford algebras whose generators satisfy the anticommutation relations, there are also the algebras whose generators satisfy commutation relations, such that the commutators of two generators are equal to the components of a metric, which is now antisymmetric. The Clifford algebras with a symmetric metric are called orthogonal Clifford algebras, whereas the Clifford algebras with an antisymmetric metric are called symplectic Clifford algebras [13]. We will see that symplectic basis vectors are in fact quantum mechanical operators of bosons [14, 15]. The Poisson brackets of two classical phase space coordinates are equal to the commutators of two operators. This is so, because the Poisson bracket consists of the derivative and the symplectic metric which is equal to the commutator of two symplectic basis vectors. The derivative acting on phase space coordinates yields the Kronecker delta and thus eliminates them from the expression. What remains is the commutator of the basis vectors. Similarly, the basis vectors of an orthogonal Clifford algebra are quantum mechanical operators for fermions. This becomes evident in the new basis, the so called Witt basis. By using the latter basis vectors and their products, one can construct spinors. Orthogonal and symplectic Clifford algebras can be extended to infinite dimensional spaces [14, 15]. The generators of those infinite dimensional Clifford algebras are fermionic and bosonic field operators. In the case of fermions, a possible vacuum state can be the product of an infinite sequence of the operators [14, 15]. If we act on such a vacuum with an operator that does not belong to the set of operators forming that vacuum, we obtain a \hole" in the vacuum. This hole behaves as a particle. The concept of the Dirac sea, which is nowadays considered as obsolete, is revived within the field theories based on Clifford algebras. But in the latter theories we do not have only one vacuum, but many possible vacuums. This brings new possibilities for further development of quantum field theories and grand unification. Because the generators of Clifford algebras are basis vectors on the one hand, and field operators on the other hand, this opens a bridge towards quantum gravity. Namely, the expec- tation values of the “flat space" operators with respect to suitable quantum states composed of many fermions or bosons, can give \curved space" vectors, tangent to a manifold with non vanishing curvature. This observation paves the road to quantum gravity. 2Here we did not go into the mathematical subtleties that become acute when the Clifford space is not flat but curved. Then, strictly speaking, the Clifford space is a manifold, such that the tangent space in any of its points is a Clifford algebra. If Clifford space is flat, then it is isomorphic to a Clifford algebra. 2 2 Clifford space as an extension of spacetime Let us consider a flat space M whose points are possible positions of the center of mass P of a physical object O. If the object's size is small in comparison to the distances to surrounding objects, then we can approximate the object with a point particle. The squared distance between two possible positions, with coordinates xµ and xµ + ∆xµ, is 2 µ ν ∆s = ∆x gµν∆x : (1) Here index µ runs over dimensions of the space M, and gµν is the metric tensor. For instance, in the case in which M is spacetime, µ = 0; 1; 2; 3, and gµν = ηµν = diag(1; −1; −1; −1) is the Minkowski metric. The object O is then assumed to be extended in spacetime, i.e., to have an extension in a 3D space and in the direction x0 that we call \time". There are two possible ways of taking the square root of ∆s2. Case I. p µ ν ∆s = ∆x gµν∆x (2) Case II. µ ∆x = ∆x γµ (3) In Case I, the square root is a scalar, i.e., the distance ∆s. In Case II, the square root is a vector ∆x, expanded in term of the basis vectors γµ, satisfying the relations 1 γ · γ ≡ (γ γ + γ γ ) = g : (4) µ ν 2 µ ν ν ν µν µ µ µ µ µ If we write ∆x = ∆x γµ = (x − x0 )γµ and take x0 = 0, we obtain x = x γµ, which is the position vector of the object's O center of mass point P (Fig. 1), with xµ being the coordinates of the point P . µ Figure 1: The center of mass point P of an extended object O is described by a vector x γµ. 3 In spite of being extended in spacetime and having many (practically infinitely many) degrees of freedom, we can describe our object O by only four coordinates xµ, µ the components of a vector x = x γµ. The γµ satisfying the anticommutation relations (4) are generators of the Clifford algebra Cl(1; 3). A generic element of Cl(1; 3) is a superposition 1 1 1 X = σ1 + xµγ + xµνγ ^ γ + xµνργ ^ γ ^ γ + xµνρσγ ^ γ ^ γ ^ γ ; (5) µ 2! µ ν 3! µ ν ρ 4! µ ν ρ σ where γµ ^ γν, γµ ^ γν ^ γρ and γµ ^ γν ^ γρ ^ γσ are the antisymmetrized products γµγν, γµγνγρ, and γµγνγργσ, respectively. They represent basis bivectors, 3-vectors and 4-vectors, respectively. The terms in Eq. (5) describe a scalar, an oriented line, area, 3-volume and 4-volume. The antisymmetrized product of five gammas vanishes identically in four dimensions. A question now arises as to whether the object X of Eq. (5) can describe an µ extended object in spacetime M4. We have seen that x = x γµ describes the centre of 1 µν mass position. We anticipate that 2! x γµ ^ γν describes an oriented area associated with the extended object. Suppose that our object O is a closed string. At first approximation its is described just by its center of mass coordinates (Fig. 2a). At a better approximation it is described by the quantities xµν, which are the projections of the oriented area, enclosed by the string, onto the coordinate planes (Fig. 2b). If we probe the string at a better resolution, we might find that it is not exactly a string, but a closed membrane (Fig. 3). The oriented volume, enclosed by this 2- dimensional membrane is described by the quantities Xµνρ. At even better resolution we could eventually see that our object O is in fact a closed 3-dimensional membrane, enclosing a 4-volume, described by xµνρσ. Our object O has finite extension in the 4-dimensional spacetime. It is like an instanton. 3 x 23 13 X µ X M4 X x2 1 Xµ(ξ) x X12 (a) (b) Figure 2: With a closed string one can associate the center of mass coordinates (a), and the area coordinates (b)). Let us now introduce a more compact notation by writing 4 X µ1µ2...µr M X = x γµ1µ2...µr ≡ x γM ; (6) r=0 4 µ M4 X X123 Xµ(ξ) Figure 3: Looking with a sufficient resolution one can detect eventual presence of volume degrees of freedom.

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