Quantum mechanics in the geometry of space time; elementary theory

The Real Geometrical Algebra or Space–Time Algebra

The Clifford Algebra Associated with an Euclidean Space

The physicists construct their experiments in a particular galilean frame {e μ }, the laboratory frame The objects and the equations expressing a theory are written in this frame.

The laws of Nature operate independently of any Galilean frame, with entities related to particles defined autonomously The crucial aspect is the Lorentz rotation, which facilitates the formulation of these laws While matrices are commonly utilized, they complicate the process compared to using two specific algebras Notably, certain elements of the Grassmann algebra of M correspond to real physical objects, such as the proper angular momentum and the bivector spin of the electron, which hold significant physical importance.

In the language of the complex spinors, the imaginary number i =√

The value -1 in the Dirac equation for the electron represents a bivector, a tangible entity that, when subjected to Lorentz rotation and multiplied by c/2, defines the angular momentum.

The first step in the use of these objects is the writing of the vectors independently of their components on a frame Compare the writing a μ b ν −b μ a ν , called also

“anti-symetric tensor of rank two”, or simple bivector, with a∧b.

R Boudet, Quantum Mechanics in the Geometry of Space–Time, 7 SpringerBriefs in Physics, DOI: 10.1007/978-3-642-19199-2_2, © Roger Boudet 2011

8 2 The Clifford Algebra Associated with the Minkowski Space–Time M

Clifford algebras are relatively obscure, yet the Clifford algebra Cl(E) of a Euclidean space E = R^p, n − p serves as the most straightforward algebra for examining the characteristics of the orthogonal group O(E) of E.

1 The definition and properties of the Grassmann algebra∧R n , of the p-vectors, elements of∧ p R n , and of the inner product between a p-vector and a q-vector in an euclidean space E =R p , n − p , are recalled in detail inChap 14.

A p-vector of E is essentially an antisymmetric tensor of rank p, represented by the components of the vectors in a frame of E that define it However, utilizing p-vectors does not require reliance on a specific frame, as demonstrated below.

In this article, we define the inner products of a p-vector \( A^p \) with a vector \( a \) in the space \( E \), a process referred to by physicists as "contraction on the indices." Additionally, the product \( a \tilde{b} \) (where \( a, b \in E \)) establishes the signature \( R^{p, n - p} \) of the space \( E \).

We will use in particular the relation

(a∧b)ãc=(bãc)a−(aãc)b, a,b,c∈E (2.1) which defines a vector orthogonal to c, situated in the plane (a,b).It will be employed for the definition of a bivector of rotation.The relation

(Bãc)ãd =Bã(c∧d)∈ R, c,d ∈ E, B∈ ∧ 2 E (2.2) will be also used.

2 The Clifford algebra Cl(E) associated with an euclidean space E is a real associative algebra, generated byRand the vectors of E, whose elements may be identified to the ones of the Grassmann algebra∧E.Furthermore this algebra implies the use of the inner products in E.

The Clifford product of two elements A,B of Cl(E)is denoted A B and verifies the fundamental relation a 2 =aãa∈R, ∀a∈E (2.3)

We simply mention in this chapter the properties we need, the complements lie inChap 14.

If p vectors a i ∈E are orthogonal their Clifford product verifies a 1 a p =a 1∧ ∧a p , (a k ∈ E, a i ãa j =0 if i= j) (2.4)

In particular a 1 ,a 2∈ E, a 1ãa 2=0⇒a 1ãa 2=a 1 a 2=a 1∧a 2= −a 2∧a 1= −a 2 a 1 (2.5)The even sub-algebra Cl + (E)of Cl(E)is composed of the sums of scalars and

2.1 The Clifford Algebra Associated with an Euclidean Space 9

From equation (2.4), it is evident that an orthonormal frame of E can be associated with the frame of ∧E, leading to the conclusion that the dimensions of Cl(E) and ∧E are both 2n, while the dimension of Cl+(E) is 2n - 1.

One uses the following operation called “principal anti automorphism”, or also

A∈Cl(E)→ ˜A∈Cl(E)so that(A B)˜ = ˜BA˜ λ˜ =λ, a˜=a, λ∈R, a∈ E (2.6)

The Clifford Algebras and the ‘‘Imaginary Number’’ ffiffiffiffiffiffiffi

Let{e 1 , e 2}be a positive orthonormal frame ofR 2 , 0 We can write

So a square root of−1 may be interpreted like a bivector ofR 2 , 0 , a real object!

The Clifford algebra Cl + (2,0) can be associated with the field of complex numbers, denoted as C Importantly, the geometric interpretation of C, especially in defining rotations in the plane formed by the basis vectors e1 and e2, predates the development of Clifford algebras For instance, utilizing the equation (2.1) provides insights into this relationship.

( e 2∧e 1 )ãe 1=e 2 , ( e 2∧e 1 )ãe 2= −e 1 which corresponds to the rotation of the vector of the frame through an angle of π/2.In Cl(M)this relation may be written i e 1=e 2 e 2 1 =e 2 , i e 2=(−e 1 e 2 ) e 2= −e 1 e 2 2 = −e 1 , i =e 2 e 1

Let{e μ }a positive orthonormal frame, or galilean frame, of M be We can write also

In the Dirac equation for a spin-up electron, the "number" i represents the bivector e 2∧e 1, while for a spin-down electron, it is replaced by −i.

The bivector spin of the electron, represented as (c/2)(n2 ∧ n1), becomes a pure real geometrical object after applying a Lorentz rotation and multiplying by c/2, rendering the equation independent of any Galilean frame Additionally, by employing a similar approach, one can identify another significant geometrical object in Cl(M), where its square is also equal to -1, highlighting its distinct role as a 4-vector that remains independent of all orthonormal frames, whether fixed or moving, within the manifold M.

10 2 The Clifford Algebra Associated with the Minkowski Space–Time M i=e 0∧e 1∧e 2∧e 3=e 0 e 1 e 2 e 3∈ ∧ 4 M, so that i 2 = −1 (2.9)

It corresponds in physics to the i of the writing F =E+i H of the bivector electromagnetic field F∈ ∧ 2 M.

So two quite different real geometrical objects, playing a fundamental role in the particles theories, are represented in the complex language by the same “imaginary number” i!

Let us denote e k =e k ∧e 0=e k e 0 , k=1,2,3 (2.10) Applying (2.2), (2.1) one deduces e k ãe j =(e k ∧e 0 )ã(e j ∧e 0 )= −e k ãe j

−e k ãe j =0 if k = j, −e k ãe k =1, k,j =1,2,3 and so these bivectors of M may be considered as a frame of a space E 3 (e 0 )=R 3 , 0 , and also, as it easy to establish by using e k =e k e 0, i =e 0 e 1 e 2 e 3=e 1 e 2 e 3=e 1∧e 2∧e 3∈ ∧ 3 E 3 (e 0 ), i 2 = −1 (2.11)

Since the e k and the i e k may be considered as bivectors ofR 1 , 3 one deduces that

Cl + (1,3)may be identified with the ring of the Clifford biquaternions Cl(3,0).The writing F =E+i H in E 3 (e 0 )of F ∈ ∧ 2 M corresponds to the definitions(2.10), (2.11).

The Field of the Hamilton Quaternions and the Ring

Hamilton introduced in its theory of the quaternions three objects i,j,k whose square is equal to−1, so that a quaternion q is in the form q=d+i a+j b+kc, a,b,c,d ∈R, i 2 = j 2 =k 2 = −1 (2.12) verifying i = −j k, j = −ki, k= −i j (2.13)

It was the first example of the fact that different objects are such that their square is equal to−1.

In fact i,j,k may be written in the form (with a change of sign with respect to the initial presentation by Hamilton)

2.3 The Field of the Hamilton Quaternions 11 k=e 1∧e 2=e 1 e 2=i e 3 (2.14) and their squares in Cl(3,0)is equal to−1, in such a way that one can write q ∈

Furthermore (2.13) may be deduced from this interpretation For example i = −j k= −( e 3 e 1 )( e 1 e 2 )=e 2 e 3

One can write q=d+i a∈Cl + (3,0) with d∈R, i a∈ ∧ 2 E 3 (e 0 ) (2.15) The biquaternions may be written

Q=q 1+i q 2∈Cl(3,0), q 1 ,q 2∈Cl + (3,0) (2.16) also as a consequence of (2.10), (2.11)

The Hamilton quaternions, specifically the Cl + (3,0) field, represent a unique algebraic structure associated with Euclidean spaces for dimensions greater than two This field is significant in the study of hydrogenic atoms, highlighting its importance in theoretical physics.

Completing a sentence of the philosopher Kant one can say “The three-space in which we live is a certitude algebraically apodictic”.

The ring Cl(3,0) serves as the algebraic continuation of the field Cl+(3,0), which corresponds to Cl+(1,3) in the context of the Minkowski space-time signature (1,3) This relationship highlights a significant alignment between human cognitive constructs and the fundamental laws of Nature Additionally, the Dirac wave function, essential for electron theory and the study of elementary particles such as quarks and leptons, is represented as an element of the privileged ring Cl+(1,3) or Cl(3,0) when expressed in real terms.

3 A Sommerfeld, Atombau und Spectrallinien (Fried Vieweg, Braunschweig, 1960)

5 R Boudet, Relativistic Transitions in the Hydrogenic Atoms (Springer, Berlin, 2009)

6 D Hestenes, Space–Time Algebra (Gordon and Breach, New-York, 1966)

Comparison Between the Real and the Complex Language

Abstract This Chapter is devoted to the definition of the Hestenes spinor and its comparison with the Dirac spinor.

Keywords Lorentz rotationãTakabayasi angleãMoving frame

The Space–Time Algebra and the Wave Function

Hestenes has shown that the wave function of an electron, when analyzed in a Galilean frame, can be represented as a biquaternion element ψ of Cl + (M) This can be expressed in the form ψ = √ρ e^(i β/2) R, where ρ is greater than zero, β is a real number, and R is a specific operator satisfying the relation RR˜ = ˜R R = 1, with i being an element of the algebra and i² = -1.

In fact one can write ψψ˜ ∈Cl + (M)⇒ψψ˜ =λ+B+iμ, λ, μ∈ R, B∈ ∧ 2 M and from(ψψ)˜ ˜ =ψψ,˜ B˜ = −B, ˜i =i,we deduce B=0 and ψψ˜ =λ+iμ=ρe i β , ψψ˜ ρe i β =1, R= ψ

So R verifiesR˜ =R − 1 and corresponds to a representation of S O + (M)in Cl + (M), that is a Lorentz rotation.

In Eq 2.9, the definition of i is presented, where the scalar ρ represents the invariant probability density The angle β is not relevant to the subsequent analysis and can be omitted in the formulation of the following currents.

14 3 Comparison Between the Real and the Complex Language

Since ai = −i a if a ∈M,a exp(iβ/2)=(exp(−iβ/2)a, and a∈ M⇒b=RaR˜∈ M, ψaψ˜ =ρb∈ M one can write ψe 0 ψ˜ =ρv= j, v 2 =1, ρ >0 (3.2)

The time-like vector j is the probability current.

Three other currents may be defined ψe k ψ˜ = j k =ρn k , n 2 k = −1, k=1,2,3 (3.3)

The vectors n k are crucial in electron theory and are linked to three bosons within the electroweak theory, as well as in our proposed framework for chromodynamics theory.

In reference [2], an expression akin to Eq 3.1 is presented using Dirac matrices, which introduces additional complexities in the application of the Dirac spinor ψ.

We shall call the biquaternionψa Hestenes spinor when it is written in the form of Eq.3.1and applied to the study of quantum mechanics.

Given all the applications in Physics of Cl + (1,3), Hestenes has given to this ring the name of Space–Time Algebra (STA) [3].

In gauge theories, the density ρ is not involved; instead, the vectors n₁ and n₂ in the U(1) gauge, along with the three vectors nₖ in the SU(2) gauge, are significant These vectors also contribute to the definition of momentum-energy tensors, as discussed in Sections 5.2 and 8.2.2.

Note on the “angle” β The role of the Yvon–Takabayasi–Hestenes “angle”β, which concerns not the vectors but the bivectors of M is obscure.

This perspective offers a more satisfactory interpretation of the electron-positron connection than the traditional approach of utilizing the T transformation of CPT invariance when transitioning from the electron's equation to that of its corresponding positron.

We show in Sect 7.2 that this interpretation proposed by Takabayasi (see [4],

Eq 10.3 b ) is in fact a necessity.

The scalar β, initially introduced in electron theory by Yvon, has been utilized in Lochak's expression of the electron's wave function This concept was further explored in depth by Takabayasi and was independently rediscovered by Hestenes.

The inclusion of the scalar β in physical theories is considered "strange" by Louis de Broglie, especially in relation to the commonly used probability ρ and Lorentz rotation R in standard quantum mechanics However, β is a crucial component of a biquaternion within Cl+(1,3), and its necessity is validated by the role biquaternions play in physics Notably, biquaternions were first introduced by Sommerfeld in the analysis of hydrogenic atoms.

We have shown in [7] thatβhas a value non null, though small, in the solution of

3.1 The Space–Time Algebra and the Wave Function 15

One can give to this entity a geometrical interpretation by considering G exp(iβ/2)R as defining a group G of transformations X →G XG (see [8]) that we˜ have called the Hestenes group.

In the context of transformations, the equation Becausei˜ = i, ai = −i a, where a and a are elements of M, indicates that this group simplifies to the Lorentz rotations when X is a vector However, when X is a bivector, it represents a type of "rotation." Additionally, the transformation β → β + π enables the inversion of the orientation of a simple bivector.

The "angle" involved in the transition from a particle to its antiparticle is significant; however, it does not affect gauge theories, as these theories pertain only to the rotations of sub-frames of M and consequently involve only sub-groups.

The Takabayasi–Hestenes Moving Frame

The role of the vector j , and sov[given by Eq.3.2in STA] in the theory of a particle whose wave function is a Dirac spinor, is well known.

In standard particle theory, the significance of the vectors \( n_k \) is often overlooked, despite the fact that the bivector spin of the electron can be expressed as \( (c/2)n_2 \wedge n_1 \) for "up" and \( (c/2)n_1 \wedge n_2 \) for "down." However, the Louis de Broglie school in the 1950s made notable contributions by incorporating these vectors into their studies of the electron, particularly through the introduction of a local orthonormal frame.

F = {v,n 1 ,n 2 ,n 3}, v=Re 0 R,˜ n k =Re k R˜ (3.4) called by Habwachs [9] the Takabayasi moving frame and considered independently by Hestenes in [1].

The vectors \( n_k \) are essential for defining invariant entities, gauge theories, and the energy-momentum tensors The finite rotations of the sub-frames \( (n_1, n_2) \) and \( (n_1, n_2, n_3) \) are closely linked to U(1) and SU(2) gauge theories, with their infinitesimal rotations contributing to the formulation of momentum-energy tensors Understanding the geometric nature of energy in particle theories necessitates the consideration of these sub-frames.

Equivalences Between the Hestenes and the Dirac Spinors

In addition to γ μ ⇔ e μ , justified in Chap 15by Eq 15.9, one can deduce the equivalences, not at all evident (see Sect 15.4), established for the first time by Hestenes [1],

The Dirac spinor can be viewed as a column of four complex numbers that interact with Dirac matrices, while Q represents a biquaternion element This comparison highlights the relationship between real and complex language in the context of advanced mathematical frameworks.

Furthermore, in the theories of the spin 1/2 particles one uses the equivalence i = i ⇔ψe 2 e 1=ψi e 3 e 0=ψi e 3 , i =√

−1⇔e 2 e 1=i e 3 (3.6) whereψis a Hestenes spinor, which is in addition to Eq.3.5the key of the translation in STA of the Dirac spinor in the theory of the electron.

The transformation of the variable 'i' into 'e' corresponds to the previously established change of 'i' into 'γ', as utilized by Sommerfeld and Lochak This relationship is reflected in the formulation of the Dirac spinor, where the 'γ μ' implicitly corresponds to the 'e μ'.

In the theories implying the SU(2) gauge we have replaced the equivalence

The Dirac current j ∈M associated with a Dirac spinor is given by the equivalence (seeSect 15.4) j μ = ¯ γ μ ∈R ⇔ j = j μ e μ =ψe 0 ψ˜ ∈M (3.8)

Comparison Between the Dirac and the Hestenes Spinors

In quantum mechanics, it is essential to transform a theory initially expressed in a Galilean frame into a form that remains invariant across all Galilean frames This invariant form is vital for providing a clear interpretation of the theory's terms in relation to space-time.

So the Lorentz rotation R which allows this passage plays a fundamental role.

The Dirac spinor, consisting of a column of four complex numbers, does not inherently include a Lorentz rotation To incorporate such a rotation, Dirac matrices must be utilized These matrices facilitate the calculation of the probability density current \( j \), the invariant density, and the corresponding unit vector \( v \).

But the calculation of the other unit vectors is much more difficult with the use of the Dirac spinors.

The determination of the vectors n k is obtained by one line in STA, that is

Eq.3.3, followed by the division byρ j 2

2 G Jakobi, G Lochak, C.R Ac Sc (Paris) 243, 234 (1956)

4 T Takabayasi, Supp Prog Theor Phys., 4, 1 (1957)

5 J Yvon, J Phys et le Radium VIII, 18 (1940)

6 A Sommerfeld “Atombau und spectrallinien” (Fried Vieweg, Braunschweig, 1960)

8 R Boudet, in “Clifford Algebras and their Applications in Mathematical Physics”, A Micali,

R Boudet, J Helmstetter, (eds.) (Kluwer, Dordrecht, 1992) p 343

9 F Halbwachs, Théorie relativiste de fluides à spin (Gauthier-Villars, Paris, 1960)

The U(1) Gauge in the Complex and Real Languages GeometricalProperties and Relation with the Spin and the Energy of a Particle of Spin 1/2

Geometrical Properties of the U(1) Gauge

This article discusses the concept of gauge change and its implications for gauge invariance, focusing on both the wave function of a particle and the potential vector influencing it The analysis is presented in a complex language framework and is further established in a real-number context.

Keywords U(1)ã SO + (1, 3) ã Finite ãInfinitesimal rotationsã Energy

The Definition of the Gauge and the Invariance of a Change

of a Change of Gauge in the U(1) Gauge

4.1.1 The U(1) Gauge in Complex Language

The Dirac spinor, represented as a column of four complex numbers, is associated with a particle and serves as the foundation for the action of Dirac matrices A gauge change is characterized by a specific transformation applied to this spinor.

The numberχmay be fixed or dependent on the point x of M and in this cases the change of gauge is to be said global or local.

4.1.2 The U(1) Gauge Invariance in Complex Language

Let us consider, associated to a particle submitted to a potential A∈ M,an expression in the form

22 4 Geometrical Properties of the U(1) Gauge

L μ =∂ μ i−g A μ , (4.3) where is a Dirac spinor expressing the wave function of the particle and g a suitable physical constant.

In the Dirac theory of the electron one has g=q/(c)where q = −e,e>0,is the charge of the electron and so g A μ has the dimension of the inverse of a length.

In a change of a local gauge like Eq.4.1, L μ becomes

If L μ is chosen in such a way that L μ = L μ U,nothing is changed, except the transform of intoU,if A μ is changed into

Such a change of the potential A does not affect the field F =∂∧A associated with

4.1.3 A Paradox of the U(1) Gauge in Complex Language

The definition Eq.4.1of the U(1) gauge in complex language leads to a paradox. Since i is considered as nothing else but the number√

A change of gauge is associated with an abstract property of the wave function, indicating that the number χ lacks a geometrical interpretation However, in the context of gauge transformation, χ acquires a geometrical significance as it is related to a potential that functions as a vector in Minkowski space-time (M), thus representing a gradient within this geometric framework.

Thus a geometrical interpretation of the gauge U(1)appears as a necessity.

The U(1) Gauge in Real Language

The geometrical interpretation of the U(1) gauge as a subgroup of SO+(1,3) has been discussed in various sources, yet it remains largely unknown among physicists, with some even rejecting it vehemently This skepticism stems from misunderstandings regarding the significance of the imaginary unit i in electron theory and a lack of awareness of the relationship expressed in equation (4.6) Consequently, critics argue that since i equals i, it is impossible for U(1) to be viewed as a subgroup of SO+(1,3).

4.2 The U(1) Gauge in Real Language 23

4.2.1 The Definition of the U(1) Gauge in Real Language

The Dirac wave function being expressed in the form Eq.3.1given by Hestenes [3],

Eq.4.4transforms Eq.4.1into ψ → ψ =ψU, U=e e 2 e 1 χ/ 2 , χ ∈R (4.6) where e e 2 e 1 χ/ 2 =cos(χ/2)+sin(χ/2)e 2 e 1

Note thatψis to be multiplied on the right by U =e e 2 e 1 χ/ 2

But here, what is called a change of gauge U(1) in complex language, corresponds in STA to

U=e e 2 e 1 χ/ 2 , R→ R =RU = Re e 2 e 1 χ/ 2 (4.7) which induces a rotation through an angleχin the plane(n 2 ,n 1 ): n 2 =cosχ n 2+sinχn 1 , n 1 = −sinχn 2+cosχn 1 (4.8(1)) with

So the replacing of iχ/2 by e 2 e 1 χ/2 gives toχthe real geometrical meaning of an angle.

The plane defined by the bivector spin of a spin-1/2 particle, represented as (n2, n1) for "up" and (n1, n2) for "down," allows for a precise definition of the U(1) gauge.

The U(1)gauge is the ring of the rotations upon itself of the plane defined by the bivector spin of a particle of spin 1/2.

4.2.2 The U(1) Gauge Invariance in Real Language

−1 by the bivector e 2∧e 1=e 2 e 1in Eq.4.3gives

In this context, it is essential to write e² e¹ to the right of ψ, allowing for the equation i = i Despite this distinction, the calculations remain consistent with those in Section 4.1.2, ultimately leading to Equation 4.5, where the previously mentioned paradox is resolved.

24 4 Geometrical Properties of the U(1) Gauge

A change of gauge through an angleχimplies ω→ω =ω−∂χ ∈M (4.10) where ω μ =∂ μ n 2 n 1= −∂ μ n 1 n 2 (4.11) which expresses the infinitesimal rotation upon itself of the plane(n 2 ,n 1 ).

The invariance is achieved by the change of the potential A

2g (4.12) as in Eq.4.5but with a meaning ofχgiven by Eqs.4.8,4.10,4.11, whose meanings are purely geometrical.

The relationship between the vector ω and the particle's energy is indicated by the change in potential, suggesting that ω, when multiplied by an appropriate physical constant (c/2 for electrons), is connected to the energy of the particle.

It is a first indication of the role played by the infinitesimal rotation upon itself of the plane(n 2 ,n 1 )in the geometrical interpretation of the energy associated with the particle.

As established inSects 5.3and6.6for the electron we will be able to insure (see [4]) that

The infinitesimal rotation of the plane defined by a particle's bivector spin, specifically for a spin 1/2 particle like the electron, determines the particle's energy when multiplied by an appropriate physical constant, such as c/2.

1 G.G Jakobi, G Lochak, C R Acad Sci (Paris) 243, 234 (1956)

2 F Halbwachs, Théorie relativiste de fluides á spin (Gauthier-Villars, Paris, 1960)

Relation Between the U(1) Gauge, the Spin and the Energy of a Particle of Spin 1/2

Abstract Real language allows one to put in evidence the relation between the U(1) Gauge, the Spin and the Energy of a particle of spin 1/2.

Keywords Spin planeãInfinitesimal rotationãMomentum-energy

Relation Between the U(1) Gauge and the Bivector Spin

In the theory of the electron, the plane(n 1 ,n 2 )has been called by Hestenes [1], the

“spin plane” because the bivector spin, or proper angular momentum, of the electron is in the form “spin up”, σ =c

In “spin down” n 2∧n 1is replaced by n 1∧n 2.

The relations between the U(1)gauge and the bivector spin of the electron, clearly established in [2] and discovered independently in [1] is absent in the standard use of the complex formalism.

Relation Between the U(1) Gauge and the Momentum–Energy Tensor Associated with the Particle

Momentum–Energy Tensor Associated with the Particle

The momentum–energy tensor associated with a particle in the U(1)gauge implies, for its valuesv,n 1 ,n 2, a linear application from M in M in the form (seeChap 16) n∈ M→ N(n)=( μ ã(i(s∧n)))e μ (5.2)

The relationship between U(1) gauge symmetry, spin, and particle energy can be expressed through the equation μ = 2(∂μ R)R - 1, where s = n^3 = Re^3 R - 1 This formulation is further multiplied by ρg1, with g1 representing a relevant physical constant, which equals c/2 for electrons.

N(n 1 )=(∂ μ n 2ãv)e μ , N(n 2 )= −(∂ μ n 1ãv)e μ (5.4) which expresses the infinitesimal rotation of the sub-frame{v,n 2 ,n 1}upon itself but in such a way that a change of gauge only affects the infinitesimal rotation of the plane(n 2 ,n 1 ).

Geometrical Properties of the Dirac Theory

The Hestenes Real form of the Dirac Equation

To eliminate any confusion regarding the electron's charge in the context of the Dirac equation, we define the charge of the electron as q = −e, where e is a positive constant (e > 0), as referenced in sources [1] and [2].

One can pass immediatly from the Dirac equation in the galilean frame{e μ } cγ μ ∂ μ (i)−mc 2 −q A μ γ μ =0, i =√

−1, q= −e, (e>0) (6.1) where∂ μ =∂/∂x μ ,to the form given to this equation in ([3], Eq 2.15), ce μ ∂ μ ψe 2 e 1 e 0−mc 2 ψ−q Aψe 0=0, A=A μ e μ ∈M (6.2) by using Eqs.3.6, then3.5, with Q =∂ μ ψe 2 e 1 Note that e 2 e 1 e 0may be written i e 3

Multiplying on the right by e 0 we obtain a form more appropriated (see Sect 7.1) ce μ ∂ μ ψe 2 e 1−mc 2 ψe 0−q Aψ=0, A= A μ e μ ∈ M (6.3) Note that each term of this equation has the dimension of an energy.

30 6 The Dirac Theory of the Electron in Real Language

In the equation ([4], Eq.5.1), the terms e 1 e 2 can be interchanged with e 2 e 1, representing two distinct states of the electron: "up" (e 2 e 1) and "down" (e 1 e 2) These states are associated with the orientation of the bivector spin.

We will work only with Eq.6.3, the second equation giving results which may be easily deduced from the change of e 2 e 1into e 1 e 2

The Probability Current

The probability current is given by Eq.3.2 It is used with the following normalization ψe 0 ψ˜ =ρv= j, ρ >0, v 2 =1, j 0 ( r )dτ =1, j 0 = jãe 0 (6.4) where the integration of j 0 is made in all the three-space E 3 (e 0 ).

Conservation of the Probability Current

The probability current j verifies the relation

∂ã j = [e μ ∂ μ (ψe 0 ψ)]˜ S = [e μ (∂ μ ψ)e 0 ψ]˜ S + [ψe 0 (∂ μ ψ)e˜ μ ] S =I+ ˜I where[X] S means the scalar part of X,and the relation[e μ Y] S = [Y e μ ] S has been applied.

So one can deduce from Eq.6.2after multiplication on the right by e 1 e 2 ψ˜

I = ρ c[mc 2 e i β n 1 n 2+qi An 3] S wherevn 1 n 2= −i n 3 , Ai = −i A has been used So I is null, as is its reversionI˜, because no term of the right hand of this equation is a scalar.

The Proper (Bivector Spin) and the Total Angular–Momenta

6.4 The Proper (Bivector Spin) and the Total Angular–Momenta

2 n 2∧n 1 (6.6) is easily calculated as explained inSect 3.3.

It defines a plane(n 2 ,n 1 )which was called, we recall, by Hestenes [4] the “spin plane”.

The total angular–momentum is defined as

J=x∧p+σ (6.7) where p is defined by Eq.6.11.

The Tetrode Energy–Momentum Tensor

The energy–momentum (Tetrode) tensor T [5] of the Dirac electron is a linear application of M into M, written by Hestenes [3] in STA n∈ M →T(n)=c

The Hestenes form of T is justified by the correspondence Eq.6.13below concerning the trace of this tensor which lies in the lagrangian of the Dirac equation.

We have shown in [6] that

In fact, a simple calculation shows that we can write

2(ρ( μ i−∂ μ β)+i∂ μ ρ)sn from which, since in particular

[ μ i sn] S = μ ã(i(s∧n)), [i(∂ μ ρ)sn] S =0 we deduce Eq.6.9.

32 6 The Dirac Theory of the Electron in Real Language Writing

2 (N(n)−(nãs)∂β) (6.10) we have, applying Eq.5.2, the so-called energy–momentum vector p=T 0 (v)−q A= c

2 ω−q A, ω=(∂ μ n 2 n 1 )e μ (6.11) which is, as shown inSect 4.2.2, gauge invariant.

A form similar to Eq.6.10of the Tetrode tensor, including the presence of∂β, has been explicited by Halbwachs [7], but with a mechanical interpretation of N(n) different from our geometrical one.

The trace ofρT 0 (n)is ρT 0 (e ν )ãe ν =c

V ãe μ (6.12) where[X] V means the vector part of X ∈Cl(M).

So the trace of the tensor ρT 0 appears in the lagrangian of the Dirac electron following the correspondence γ¯ μ i∂ μ ⇐⇒ e μ (∂ μ ψe 2 e 1 )e 0 ψ˜

V (6.13) deduced from the two correspondences Eqs.3.6then3.5.

The presence of e 0 e 2 e 1in the lagragian is a hint on the fact that the momenum– energy tensor contains the expression of an infintesimal rotation of the sub-frame

Relation Between the Energy of the Electron and

Infinitesimal Rotation of the “Spin Plane”

As a confirmation of what we said inSect 5.3, the vector c

2 ω= p+q A, ω=(∂ μ n 2ãn 1 )e μ is such thatω 0is the energy E of the electron in the galilean frame{e μ }:

We have by a direct calculation verified in [6] this property for the hydrogen atom.

The Tetrode Theorem

The Tetrode theorem [5] is the following:

“The space–time divergence of the energy–momentum tensor of the Dirac electron is equal to the density of the Lorentz force acting on the electron".

Let us replace the vectors n by vectors e ν of the frame{e μ }.One can write

Chapter 18 contains a STA proof of the Tetrode theorem which allows one to shorten the proof given by Tetrode [5], published in 1928 (just after the article of Dirac !).

The Lagrangian of the Dirac Electron

Multiplying Eq 6.2on the right byψ˜ and taking the scalar part one has, because [ψψ]˜ S =cosβ, ρv=ψe 0 ψ˜

The Lagrangian of the Dirac electron, expressed as L = ce μ ã [(∂ μ ψ)e 0 e 2 e 1 ψ]˜ V − mc² cosβ − Aã(qρv) = 0, is fundamentally equivalent to the conventional formalism It becomes zero when the Dirac equation is fulfilled, highlighting its significance in quantum mechanics.

Units

The only constants we will use are the three fundamental constants (revised in 1989 by B N Taylor):

(1) the speed of light c=2.99792458×10 10 cm sec − 1

(3) the reduced Planck constant = h/2π = 1.054 572 × 10 − 27 erg sec.

In addition we will use

(4) the electron mass m =9.109 389×10 − 28 g All the other constants used will be derived from these four ones, in particular

137.035 989 (e in e.s.u.) (6.17) and as unit of length:

(6) the “radius of first Bohr orbit” a= 2 /(me 2 )=/(mcα)=5.291 772×10 − 9 cm (6.18)

34 6 The Dirac Theory of the Electron in Real Language

In the expression of electromagnetic potentials, a factor of 1/(4π ε₀) is introduced, where ε₀ represents the permittivity of free space This factor arises from using 4πjμ instead of jμ in the current term of Maxwell's equations.

0=8.854187×10 − 12 F m − 1 , e=1.602 1777×10 − 19 (e.m.u.) That gives (with c expressed in metres) the same value ofαwith the expression α= e 2

To ensure clarity and align with the majority of referenced articles and treatises, we will prioritize using the traditional expressions of potentials and the constant α instead of their symbolic representations.

1 L.D Landau, E.M Lifshiftz, Quantum Mechanichs, vol 4 ((Pergamon Press), New York, 1971)

2 F Halzen, D Martin, Quarks and Leptons (Wiley, USA, 1984)

7 F Halbwachs, Théorie relativiste de fluides á spin (Gauthier-Villars, Paris, 1960)

The Invariant Form of the Dirac Equation and Invariant Properties of the Dirac Theory

Abstract This section is relative to the invariant form of the Dirac equation and some fundamental invariant properties of the Dirac theory which may be deduced from this form.

Keywords Invarianceã Positron ã Broglie ã Lorentz ãEinstein formulas

The Invariant Form of the Dirac Equation

Multiplying on the right Eq.6.3first by e 2 e 1then byψ − 1 : ce μ ∂ μ ψψ − 1 = −(mc 2 ψe 0+q Aψ)e 2 e 1 ψ − 1 (7.1) whereψ − 1 = R − 1 exp(−iβ/2)/√ρ,we have the following invariant form of the Dirac equation [1] c

This equation corresponds to the state “spin up” For the state “spin down”,σ 0is to be changed into−σ 0

We recall that each bivector μ represents the infinitesimal rotation of the

“Takabayasi–Hestenes proper frame”{v,n 1 ,n 2 ,n 3}when the point x moves in the e μ direction.

This equation may be divided into two parts:

1 The∧ 3 M part D I ,four real equations implying seven real scalars R, β,is independent ofρ.These scalars, associated with the physical constants,c,q and the potential A,lead to the construction of all the entities (energy, spin) which are observable Note that D I is gauge invariant.

36 7 The Invariant Form of the Dirac Equation and Dirac Theory

2 The vector part D I I ,four equations implying eight real scalars R, β, ρ,implies in addition the density ρ which has a probabilistic (or, following the authors, statistical) meaning.

The Dirac theory, along with other physical theories, can be articulated through equations that involve observable physical entities However, these equations often include an excess of real parameters compared to the number of equations available, necessitating the introduction of additional equations that incorporate probabilistic or statistical parameters.

About the link between equations D I and D I I we have established in [2] the following theorem:

D I I is implied by D I and the three conservation relations

∂ μ (ρv μ )=0, ∂ μ (T μν )=ρf ν , ∂ μ (S μνξ )=(T ξν −T νξ ) where T is the Tetrode tensor, f ∈ M the Lorentz force and S=ρv∧σ.

In particular cases of the choice of the potential A, particular solutions of the equation D I , may lead to the expression of phenomena directly observable (seeSects 7.4,7.5).

The Passage from the Equation of the Electron to the

to the One of the Positron

The invariance conditions of the Dirac equation, particularly concerning the positron linked to an electron with a specified spin orientation, are established through standard operations in the Dirac equation framework These conditions are derived from the application of CPT transformations.

It is easy to obtain the CPT invariance by using Eq.7.2.

(b) P (Parity) changes(e 2 ,e 1 )into(e 1 ,e 2 )and so n 2 n 1into n 1 n 2

(c) T (Time reversion) changes e 0into−e 0and sovin−v.

The left hand part of Eq.7.2is unchanged by (a), (b) and (c):

– As a consequence of (b),σ 0is changed into−σ 0 ,so−q(−σ 0 ) =qσ 0and the charge term in Eq.7.2is unchanged.

– As a consequence of (c), −v(−σ 0 ) = vσ 0, and the mass term in Eq 7.2 is unchanged.

So the right hand part of Eq.7.2is unchanged The left hand part is unchanged by any of these transforms.

But the T transformation seems imply that the positrons come from the future.

In order to explaining this particularity of the T transformation, where the

7.2 The Passage from the Equation of the Electron to the One of the Positron 37

In 1948, Feynman introduced an interpretation of the T transformation, suggesting that a negative entity traveling backward in time is equivalent to a positive energy antiparticle moving forward in time This concept is rooted in the Dirac equation, which relates velocity (v) to the negative charge of the electron (−e < 0) and connects the negative velocity (−v) to the positive charge of the positron (e > 0).

In [4], Eq 10.3 b ,Takabayasi avoids the change otv into−v by the following transform:

(c) The angleβis changed intoβ+π,vremaining unchanged.

As an additional justification of the Takabayasi transformation, one can remark that:

1 The “angle”βconcerns the “rotation” of bivectors, not of vectors, and its change into β +π, implying the reversal of a bivector, is coherent with the change n 2 n 1 = n 2∧n 1 into n 1 n 2 = n 1∧n 2implied by the P transformation, which associates to an electron a positron whose the orientation of the bivector spin is opposite.

2 The angleβappears in the mass term of the lagragian of the positron in the form

The equation mc² cosβ, where 0 ≤ β ≤ π, suggests that transforming β into β + π correlates with the change of mass for the electron into a negative mass for the positron This transformation may be linked to the phenomenon of mass disappearance during the electron-positron annihilation process.

The Free Dirac Electron, the Frequency and the Clock

and the Clock of L de Broglie

The equation of the free Dirac electron may be deduced from the equation D I simply by supposing that A=0 and furthermore thatβ =0.

Multiplying Eq.7.2on the left byσ 0=n 2 n 1 ,then taking the vector part of this new equation, we have

Considering the galilean frame where the electron is at rest, we can writev =e 0 , and furthermore x 0 =ct gives the proper time t of the free electron.

So the energy of the free electron is

38 7 The Invariant Form of the Dirac Equation and Dirac Theory The introduction of the L de Broglie frequency ν 0=mc 2 h = 1

2 , ω=(∂ μ n 2 n 1 )e μ (7.5) which is so related to the infintesimal rotation of the spin plane upon itself, allows us to give a geometrical picture of what is called the L de B clock.

Let us denote n 1=cosϕ e 1+sinϕe 2 , n 2= −sinϕ e 1+cosϕ e 2 (7.6) which corresponds to R=exp(−e 2 e 1 ϕ/2).

The angleϕis only function of x 0and one deduces ω=e 0 dϕ d x 0 , cω= dϕ dtv

So we can give to the hand of the L de Broglie clock the following pure geometrical interpretation It is a vector N of the spin plane such that

N =cos(ϕ/2)e 1+sin(ϕ/2)e 2 (7.8) which runs on the direct direction of the plane(n 1 ,n 2 ),that is, on the dial of the clock, anti-clockwise.

In the case where the spin is “down”, one can see in the same way that the vector

N =cos(ϕ/2)e 1−sin(ϕ/2)e 2 (7.9) which runs on the inverse direction of the plane(n 1 ,n 2 ),that is, in the dial of the clock, clockwise.

The equation for the positron indicates that the charge q is irrelevant, allowing for the existence of an electron even when A=0 In this context, the P transformation alters Eq 7.3 by changing ω to −ω, resulting in a right-hand side of −mc²v Consequently, it is necessary to assume that v is also transformed to −v, suggesting that the positron originates from the future—a concept proposed by Stückelberg and Feynman However, this hypothesis is deemed invalid in the absence of charge in the current scenario Alternatively, one might consider that mass m is transformed to −m, consistent with the Takabayasi transformation where β = 0 and π = π.

The Dirac Electron, the Einstein Formula of the Photoeffect

7.4 The Dirac Electron, the Einstein Formula of the Photoeffect and the L de Broglie Frequency

We consider the particular, but important case that we have considered in [5], where the potential A is in the form

We can have a solution of the equation D I by assuming thatβ=0 and that the spin plane keeps a fixed direction in such a way thatσ 0=n 2∧n 1is defined by n 1=cosφe 1+sinφe 2 , n 2= −sinφe 1+cosφe 2

Since we have then A.σ 0 = 0, Aσ 0 = A∧σ 0 ,multiplying as before Eq.7.2on the left byσ 0=n 2 n 1 ,then taking the vector part of this new equation, we have the equation c

2 ω−q A=mc 2 v, ω=(∂ μ n 2 n 1 )e μ (7.11) similar to Eq.7.3, but with a potential in addition, and also deduced here from the

The time component of this equation is c

Now, assuming the approximation mc 2 v 0 =mc 2

2m v 2 (7.14) and writing mc 2 in the form h(mc 2 /h)=hν 0one deduces hν−W = 1

2m v 2 , ν=ν 1−ν 0 (7.15) that is the formula of the photoeffect introduced by Einstein in 1905.

The L de Broglie frequency, expressed as mc²/h, plays a crucial yet often overlooked role in Einstein's equation This relationship emerged following Max Planck's 1900 discovery of the energy quantum hν, which laid the groundwork for the development of quantum theory related to electrons.

40 7 The Invariant Form of the Dirac Equation and Dirac Theory

The Equation of the Lorentz Force Deduced from

from the Dirac Theory of the Electron

In this analysis, we adopt a methodology akin to that of Hestenes, as referenced in sources [6] and [7], specifically focusing on the scenario where the angle β is zero.

Taking the spacetime curl of Eq.7.11, and sinceω μ is a gradient, one obtains

We notice thatis eliminated during this operation and so we go towards a classical theory of the electron.

(∂∧v).v=(e μ ∧∂ μ v).v= −(v.∂)v one has m(Vã∂)V =q cFãV, V =cv (7.18)

Now we make the point x as describing one of the current lines C,defined in the spacetime plane(e 0 ,e 3 ),by Eq.7.11 One has along C

The equation V = d x dτ describes the relationship between velocity (V) and the proper time parameter (τ) along a curve C In this context, (V.∂)V represents the derivative of velocity with respect to the tangent vector V at point x, indicating the particle's acceleration This formulation highlights the connection between proper time and the dynamics of motion in a given trajectory.

Equation7.18becomes (see [6], Eqs 3.5) md V dτ =q cFãV, V = d x dτ (7.20) that is an equation which has exactly the same form as the Lorentz force equation and, so, may be considered as defining “a trajectory”.

However Eq.7.20is to be considered as compatible with Eq.7.11whose corresponding integral line defines a current line of the Dirac theory.

Moreover, if we consider C as a trajectory, the plane orthogonal at x to C (which keeps a fixed direction parallel to the(e 1 ,e 2 )plane), is nothing else but the “spin

7.5 The Equation of the Lorentz Force 41

In this way we can say that the Planck constant, which appears in Eq.7.11, and the spin are hidden parameters of the classical theory of the electron.

The Dirac theory is simplified to its dynamical equation, D I, while the component involving the density ρ, represented by equation D I I, has been overlooked.

In the context of spacetime, the curve C is specifically examined within a spacetime plane, where the electron's space curve is represented as a straight line.

The SU(2) Gauge and the Yang–Mills Theory in Complex

The SU(2) Gauge in the General Yang–Mills Field Theory

The Yang–Mills (Y.M.) lagrangian (see [1], p 8)

2W μ whereτ k are the isospin (or Pauli) matrices.

In this standard expression the set {τ k }is interpreted as the frame{e k }of an

“isotriplet space” isomorphic toR 3 , 0 This space is to be considered as the space

E 3 (e 0 )generated by the bivectors of M e k =e k ∧e 0=e k e 0

So the B μ appear as bivectors of M and W k =e μ W μ k as vectors of M.

The nature of the two components on which the matrices τ_k act remains unclear in most treatises, with the exception of electroweak theory This topic is explored in Section 4.1 of reference [2], as summarized in Chapter 17, concluding that the components can either be an “ordinary” Dirac spinor or a pair of Dirac spinors The only scenario in which the τ_k can be treated as matrices is when they represent a right or left doublet within the SU(2)×U(1) gauge framework, as discussed in Chapter 9.

46 The Associated Momentum–Energy Tensor

The SU(2) gauge should be interpreted not merely as entities isomorphic to bivectors of M, but as genuine geometrical bivectors, as defined by Equation 8.6, utilizing the Space-Time Algebra (STA).

The conventional gauge SU(2)transformation is achieved by the relation

B μ =U B μ U − 1 + i g(∂ μ U)U − 1 where U belongs to a sub-group of SU(2) Let us denote ˆ μ =2(∂ μ U)U − 1 (8.2) in such a way that

F νμ =∂ ν B μ −∂ μ B ν −gi(B μ B ν −B ν B μ ), F νμ =U F νμ U − 1 (8.4) or, with the qualityτ k =e k of vectors of E 3 (e 0 )(bivectors of M) given to the isospin matricesτ k and applying the relation in E 3 (e 0 )

2( ab−ba )= −i( a∧b )=a×b where a×b means the vector product in E 3 , one obtains

We remark on Eq.8.3that if W μ is a bivector, ˆ μ is a bivector too.

So the relation F νμ =U F νμ U − 1 may be deduced from∂ νμ 2 U =∂ μν 2 U by means of the definition (8.2) which gives the relation Eq.C.19.

We have chosen to use ˆ μ = 2(∂ μ U)U − 1 instead of(∂ μ U)U − 1 because it is already an indication of the geometrical meaning of the gauge.

The SU(2) Gauge and the Y.M Theory in STA

8.2.1 The SU(2) Gauge and the Gauge Invariance in STA

All that follows could be simply replaced by supposing that U is considered like an

8.2 The SU(2) Gauge and the Y.M Theory in STA 47 use of STA in detail because it brings important elements absent from the standard theory.

In the SU(2)gauge, the role of the “isocurrents” j k orthogonal to the probability current j of a particle, whose wave function is an invertible biquaternion ψ, is important.

We will use the concordances τ k ⇔e k =e k ∧e 0=e k e 0 (8.6) γ¯ μ τ k ⇔e μ ã(ψ e k e 0 ψ)˜ =e μ ã(ψe k ψ)˜ = j k μ (8.7) which will be justified by what follows.

We considerψinstead of , and we denote by U a rotation such that

The concordance with the complex L 1requires an interpretation of i(Eq.3.7), that we have introduce in [3], different from the one of Eq.3.6 Here we will write i=i ⇔iψ=ψi, i =√

Applying the concordances Eqs (8.11), (3.5) we have

2(ξ μ v+ρi μ v), ξ μ =(∂ μ ρ+iρ∂ μ β)i (8.13) where theξ μ are only function ofρ, β.

The gauge transformation represented by U in complex formalism is essentially a self-rotation of the three-dimensional orthogonal space at the point x of M, corresponding to the probability current j = ρv of the Yang-Mills particle.

We obtain the geometrical interpretation of the gauge SU(2):

The SU(2) gauge, linked to a particle, represents the rotation group in the three-dimensional space orthogonal to the particle's time-like probability current This gauge is a subgroup of SO+(1,3).

The gauge transformation leavesvinvariant but defines a rotation of the sub-frame upon{n 1 ,n 2 ,n 3}upon itself,ρ, β, and so theξ μ , remaining unchanged.

We are going to find X such that the change

W μ =U W μ U − 1 + i gˆ μ (8.16) exactly as in Eq.8.3of the complex formalism and with the same field F νμ

What is new is the geometrical interpretation of a gauge transformation in SU(2): a rotation upon itself of the three space E 3 (v).

We see the geometrical link with the gauge U(1)where the rotation is relative to the “spin plane”.

8.2.2 A Momentum–Energy Tensor Associated with the Y.M Theory

The part of a momentum–energy tensor associated with the Y.M theory which does not take into account the B μ may be written in STA n ∈M →T(n)=g 1 ∂ μ ψie 0 ψ˜n

(8.17) where g 1is a suitable physical constant.

In transitioning from a U(1) gauge to an SU(2) gauge, we replace ψi e 3 with ψi e 0, leading to the derivation of an SU(2) energy-momentum tensor This is achieved by substituting s=n 3 with v in the U(1) energy-momentum tensor, as indicated by the application of concordance (8.11) instead of (3.6) The resulting tensor, denoted as ρT, is formulated using Eq 6.10, where v replaces s, and is further expressed in terms of S(N) as per Eq 16.12.

8.2 The SU(2) Gauge and the Y.M Theory in STA 49 which expresses the infinitesimal rotation upon itself of{n 1 ,n 2 ,n 3}, replacing N(n), and a suitable physical constant g 1in place ofc n ∈M →ρT 0 (n)=ρg 1

In the gauge SU(2), the energy–momentum tensor contains the infinitesimal rotation of the three-space orthogonal to the probability current of the particle.

The trace of T(e ν )ãe ν /g 1of T(n)/g 1is

With the introduction of the physical constant g 1the Lagrangian will be written in the form

8.2.3 The STA Form of the Y.M Theory Lagrangian

We can deduce now from Eqs.8.19,8.8the equivalence

Conclusions About the SU(2) Gauge and the Y.M Theory

Considered separately the SU(2)gauge and the Y.M theory have no place in the complex language The use of STA is a necessity.

The wave function , on which the isospin matrices act, cannot be either a Dirac spinor, couple of Pauli spinors, or a couple of Dirac spinors.

The isospin matrices may be interpreted like bivectors of M, and as corresponding to a Hestenes spinorψ, invertible biquaternion.

They may be also associated with a bivector a+i bof M (see the three first lines of Table 1 of [4]).

In any case the SU(2) gauge, considered as alone, cannot be considered as deduced from complex matrices like the isospin ones, and as associated with a particle of spin 1/2.

In Section 8.1, it is noted that the τ k can be regarded as bivectors of M and U, indicating that they belong to the special orthogonal group SO+(M) rather than a Lie group.

The Y.M theory retains its properties through the complex language utilized in the SU(2) component of the electroweak theory, as well as in the section that incorporates the first three Gell-Mann matrices relevant to chromodynamics.

The first theory posits that the entity ψ is a left doublet, potentially representing an invertible biquaternion In the second theory, it is essential for ψ to be a doublet—either left or right—due to the spin 1/2 characteristic of the particles under consideration.

The SU(2) gauge theory, while not currently recognized as a distinct entity in accepted scientific theories, may still hold potential for elucidating unknown phenomena in the future.

1 M Carmeli, Kh Huleihil, E Leibowitz, Gauge Fields (World Scientific, Singapore, 1989)

2 R Boudet, Adv Appl Clifford Alg (Birkhaüser Verlag Basel, 2008), p 43

3 R Boudet, in The Theory of the Electron, ed by J Keller, Z Oziewicz (UNAM, Mexico, 1997), p 321

4 D Hestenes, Space–time structure of weak and electromagnetic interactions Found Phys 12,153–168 (1982)

The SU(2) 3 U(1) Gauge in Complex and Real Languages

Left and Right Parts of a Wave Function

In this chapter we recall the calculations achieved in [1].

We consider an “ordinary" spinor,that is a Dirac spinor which may be replaced by a Hestenes spinorψ,and so a invertible biquaternion, and its decomposition

(see [2], Eq 5.49) in the so called left and right parts of.

The equivalences deduced from Eqs.3.5,3.6 γ 5 =γ 0 γ 1 γ 2 γ 3 i⇔ e 0 e 1 e 2 e 3 ψi e 3 (e 0 ) 4 = −e 0 e 1 e 2 e 3 ψi e 3= −iψi e 3=ψ e 3

2(1∓e 3 )ψ (9.4) lead in STA to the decomposition

54 9 Geometrical Properties of the SU(2)× U(1) Gauge ψ=ψ L +ψ R (9.5) ψ L =ψ1

Let us consider the two following spacetime vectors j =ψe 0 ψ˜ =ρv, j =ψe 3 ψ˜ =ρs, s=n 3 (9.9)

We recall that the spacetime vectors j and j are respectively the probability density and the “spin density" currents of the particle, whose wave function isψ.

Note that j±j =ρ(v±s)are isotropic vectors:(j± j ) 2 =0.

2(e 3+e 0 ) (9.10) one obtains the “currents”, which have the particularity to be isotropic, j L =ψue 0 u˜ψ˜ =ψue 0 ψ˜ =1

Left and Right Doublets Associated with

We will only consider the case of the left doublets, the case of the right doublets giving similar results.

Let two ordinary Dirac spinors 1 , 2 be, in their Hestenes formψ 1 , ψ 2 ,corresponding to two particles of spin 1/2.

What follows is applicable to all the invertible biquaternions We define a left doubletψ L as ψ L =ψ L 1 −ψ L 2 e 1 (9.13) ψ ψ L 1

9.2 Left and Right Doublets Associated with Two Wave Functions 55

The choice of the vector e 1is arbitrary, all that we need is its orthogonality to e 3

Eachψ L 1 , ψ 2 L verifies the relation ψ L α e 3= −ψ α L , (α=1,2) (9.14) One can write, because e 1 2 =1,

So one can define three transformations, corresponding to the action of theτ k matrices of the conventional presentation, ψ L → −ψ L e 1 ,−ψ L e 2 ,−ψ L e 3 ⇔ L → τ 1 L , τ 2 L , τ 3 L (9.18)

Assuming that the biquaternionψ L is invertible, that isψ L ψ¯ L =0 (see a condition below), we deduce the correspondence similar to Eq.8.7, except for a change of sign, ¯ L γ μ τ k L ⇔ −e μ ã(ψ L e k e 0 ψ˜ L )= −e μ ã(ψ L e k ψ˜ L )= −j k μ (9.19)

The presence of the element e1 in the biquaternion ψL necessitates clarification The orientation of e3 enables the definition of the "spin plane" directions for both the electron and the neutrino at each point x of M, achieved through orthogonality to the vector.

The direction of e1 is arbitrarily chosen to define the plane for two independently considered particles, but a common choice is essential to analyze the correlation resulting from a U(1) gauge transformation This transformation involves a simultaneous rotation of these planes by the same angle χ.

The selection of e1 enables the translation of the aforementioned isospin matrices to STA, while opting for e2 is also permissible, albeit resulting in a different yet analogous structure of these matrices.

56 9 Geometrical Properties of the SU(2)× U(1) Gauge

9.3 The Part SU ( 2 ) of the SU ( 2 ) × U ( 1 ) Gauge

The biquaternionψ L is invertible if ψ L ψ˜ L = −(X+ ˜X)=0, X =ψ 2 u e 1 ψ˜ 1 (9.20) which is verified if X is not reduced to a bivector, that we will suppose.

Then we can consider the four currents which are not isotropic j L 0 =ψ L e 0 ψ¯ L , j L k = −ψ L e k ψ¯ L (9.21)

The orthonormal frame {j L μ} is oriented negatively in relation to the Galilean frame {e μ}, yet this alteration does not affect the properties of the SU(2) gauge as outlined in Section 8.

9.4 The Part U ( 1 ) of the SU ( 2 ) × U ( 1 ) Gauge

A U(1)change of gauge for the doublet is defined in STA by the transforms ψ 1 →ψ 1 e (± e 1 e 2 ϕ/ 2 ) , ψ 2 →ψ 2 e (± e 1 e 2 ϕ/ 2 ) (9.22)

Note that the angleϕdefining this change must be the same forψ 1 andψ 2

It is possible to deduce from e (± e 1 e 2 ϕ/ 2 ) e j e (∓ e 1 e 2 ϕ/ 2 ) =e j , j =0,3 (9.23) that the product of SU(2)by U(1)is direct.

A precise verification of this property will be achieved inSect 10.3.

9.5 Geometrical Interpretation of the SU ( 2 ) × U ( 1 ) Gauge of a Left or Right Doublet

Properties similar to the ones of the left doublet may be established for a right doublet.

We deduce from all that precedes the following properties:

1 The SU(2)XU(1)gauge may be applied onl y to a left or right doublet.

2 A change of such a product of gauges corresponds to a finite rotation in the three space orthogonal to the timelike current of the doublet and a finite rotation of a same angleϕin the “spin planes" of the particles whose wave functions areψ 1 andψ 2

9.6 The Lagrangian in the SU(2) × U (1) Gauge 57

9.6 The Lagrangian in the SU ( 2 ) × U ( 1 ) Gauge

Given that ψ L is invertible, the Lagrangian maintains the same structure as in Eq 8.21 However, due to the orientation of the three-space {−j 1, −j 2, −j 3} being the inverse of {j 1, j 2, j 3}, S(n) should be substituted with −S(n) as indicated in Eq 16.13, leading to L I being replaced by −L I in Eq 8.18.

Eq.8.19in such a way that L is changed into

Taking account this change, all that we have established inSect 8.2is still applicable to the SU(2)×U(1)gauge of the weak theory.

1 R Boudet, in The Theory of the Electron, ed by J Keller, Z 0ziewicz (Mexico, 1997), p 321

Geometrical Interpretation of the SU(2) 9 U(1) Gauge

of a Left or Right Doublet

Properties similar to the ones of the left doublet may be established for a right doublet.

We deduce from all that precedes the following properties:

1 The SU(2)XU(1)gauge may be applied onl y to a left or right doublet.

2 A change of such a product of gauges corresponds to a finite rotation in the three space orthogonal to the timelike current of the doublet and a finite rotation of a same angleϕin the “spin planes" of the particles whose wave functions areψ 1 andψ 2

9.6 The Lagrangian in the SU(2) × U (1) Gauge 57

9.6 The Lagrangian in the SU ( 2 ) × U ( 1 ) Gauge

Since the matrix ψ L is invertible, the Lagrangian retains the same structure as in Equation 8.21 However, due to the orientation of the three-space {−j 1, −j 2, −j 3} being the inverse of {j 1, j 2, j 3}, the term S(n) must be substituted with −S(n) as indicated in Equation 16.13, and LI should be replaced with −LI.

Eq.8.19in such a way that L is changed into

Taking account this change, all that we have established inSect 8.2is still applicable to the SU(2)×U(1)gauge of the weak theory.

1 R Boudet, in The Theory of the Electron, ed by J Keller, Z 0ziewicz (Mexico, 1997), p 321

The Lagrangian in the SU(2) 9 U(1) Gauge

The Glashow–Salam–Weinberg Electroweak Theory

General Approach

Hestenes has presented the electroweak theory in Spacetime Algebra (STA) through his work (see [1–3]) In contrast, our independent approach offers a more comprehensive and detailed translation of the standard presentation into STA, as outlined in references [4] and [5].

This presentation focuses exclusively on the leptonic aspects of the theory related to the first generation, specifically the electron and neutrino The extension to include hadronic currents associated with the strange and down quarks in the Cabibbo mixing, as well as the up quark, is straightforward and poses no significant challenges.

As in [6], we have chosen to present the theory in the simplest form but with the conservation of its fundamental features.

In particular we will not mention the role of entities as hypercharges, certainly important, but they do not appear in the final results.

This article presents the theory utilizing the STA, ensuring consistency with standard presentations through the alignment of entities and equations across all Galilean frames in both complex and real mathematical approaches.

62 10 The Electroweak Theory in STA: Global Presentation

The Particles and Their Wave Functions

Only two leptons are to be considered, the electron and the neutrino.

They will be considered in the “spin up” state.

Their wave functions will be expressed by the Hestenes spinorsψ e andψ ν in a galilean frame{e μ }.

10.2.1 The Right and Left Parts of the Wave Functions of the Neutrino and the Electron

This article discusses four wave functions derived from the interaction of neutrinos and electrons By applying the decomposition method outlined in Eq 9.6, we express the wave functions as ψ α = ψ α u + ψ α u˜ = ψ L α + ψ R α, where α represents either the neutrino (ν) or the electron (e) The terms are defined with u = (1 - e₃)/2 and u˜ = (1 + e₃)/2 Consequently, we define the right-handed and left-handed wave functions as ψ ν R = ψ ν u˜, ψ R e = ψ e u˜ for right-handed components, and ψ L ν = ψ ν u, ψ L e = ψ e u for left-handed components.

10.2.2 A Left Doublet and Two Singlets

Three wave functions are used.

The wave function of the left doublet in the theory is represented by the combination of the left components of the electron and neutrino, expressed as ψ L = ψ ν L - ψ L e, where e 1 = ψ ν u - ψ e u and e 1 = e 1 e 0 Additionally, the two singlet states are defined as ψ ν R and ψ e R.

The Currents Associated with the Wave Functions

Some of the following currents appear directly in the lagrangian Others may be considered as auxiliary.

10.3 The Currents Associated with the Wave Functions 63

10.3.1 The Current Associated with the Right and Left Parts of the Electron and Neutrino

Using Eqs.9.12,9.11one has for the electron j R e =1

2(j e − j e ) (10.5) that is two isotropic currents, from which one deduces j R e + j L e = j e (10.6) which is a timelike vector as in the Dirac theory of the electron.

In the same way one has for the neutrino j R ν = 1

10.3.2 The Currents Associated with the Left Doublet

The currents defined in Sect 9Eq.9.21 are expressed as j L 0 =ψ L e 0 ψ¯ L and j L k = −ψ L e k ψ¯ L, indicating they are not isotropic when Eq 9.20 is satisfied The expressions for j L 0, j L 1, j L 2, and j L 3 are derived from the relationships involving ψ and u, leading to j L 0 being simplified to 1 By recalling the relationships e k = e k e 0 and applying relevant equations, we establish the connections between the various components of the currents, reinforcing the non-isotropic nature of the system.

64 10 The Electroweak Theory in STA: Global Presentation j L 1 =1

2(j ν −j ν −(j e −j e )) (10.12) For the proof concerning j L 2 we have used e 2=i e 1 e 3 , Qi =i Q, ∀Q∈Cl + (M)

The analysis of Eq 9.23 reveals that a U(1) gauge transformation, which involves a simultaneous rotation of the spin planes for electrons and neutrinos, does not alter the left-handed current \( j_L^\mu \) This transformation, represented as \( \psi_\alpha \exp(\pm e_1 e_2 \phi/2) \) for \( \alpha = e, \nu \), remains unaffected by the SU(2) gauge change, which pertains to rotations in the three-dimensional space orthogonal to \( j_L^0 \) and involves the vectors \( j_L^k \) Consequently, it is confirmed that the combination of SU(2) and U(1) in the SU(2)×U(1) gauge framework for the left doublet is direct.

To be in agreement with the conventional presentation of the GSW theory, we introduce the so-called charged current j C = 1

2ψ e (e 0−e 3 )ψ˜ ν (10.13) and its “complex conjugate” [see Eq.10.15] j˜ C = 1

The equations presented, specifically 2ψ ν (e 0−e 3 )ψ˜ e (10.14), exemplify the V-A type, which stands for "vector-axial vector" (refer to [4], p 146) This classification is evident from Eqs 10.15 and 10.16, illustrating the combination of a vector and a pseudo-vector within the context of M A straightforward verification shows that for Q 1 , Q 2 ∈ C + (M) and a ∈ M, the relation Q 1 aQ˜2 = b + i c holds, with its conjugate given by (Q 1 aQ˜2)˜ = Q 2 aQ˜1 = b - i c, where b and c are elements of M (10.15) Additionally, these relationships can be further derived from Eqs 10.10 and 10.11.

10.3 The Currents Associated with the Wave Functions 65 j C =1

2(j L 1 −i j L 2 ) (10.16) which in agreement with the property (10.15).

The Bosons and the Physical Constants

The fundamental physical constants essential to the standard model include the speed of light (c), the electric charge (e), and the weak mixing angle (θ), with sin²θ equaling 0.234 Additionally, two other constants, g and g, are derived from e and θ, expressed in the equation g sinθ = g cosθ = e > 0 For further details on the selection of e > 0, refer to Section 6.1 and Equation 5.17 in the relevant literature.

3 Two neutral bosons W 3 ,B ∈ M and Z ∈ M, which are massive, such that ([4],

W 3 =sinθ A+cosθ Z, B =cosθ A−sinθ Z, (10.191) and in another but equivalent presentation ([5], Eqs 13.19, 13.20)

A=sinθW 3 +cosθ B, Z =cosθW 3 −sinθ B, (10.192) such that W 3 ,B, θ make the combination sinθ W 3 +cosθ B massless.

The Lagrangian

The Lagrangian is in the form

66 10 The Electroweak Theory in STA: Global Presentation

1 A part L I independent of the bosons fields (seeSect 11.1)

The equation L I = c(e μ [(∂ μ ψ ν )ie 3 ψ˜ ν ] V + e μ [(∂ μ ψ e )ie 3 ψ˜ e ] V) = L ν + L e highlights the relationship between the momentum energy tensors of neutrinos and electrons, represented as L ν and L e, respectively Notably, L e is derived from the Lagrangian in Eq 6.16 of the Dirac equation for electrons, excluding the potential term.

2 A part L I I implying the bosons fields

L I and L I I are each cut into several parts with a correspondence between each part of L I with a part of L I I

1 D Hestenes, Space–time structure of weak and electromagnetic interactions Found Phys 12, 153–168 (1982)

4 E Elbaz, De l’électromagnétisme à l’électrobaible (Ed Marketing, Paris, 1989)

6 R Boudet, in The Theory of the Electron, ed by J Keller, Z 0ziewicz, eds (Mexico, 1997), p 321

The Electroweak Theory in STA: Local

Abstract This section is relative to the standard decompositions of the theory into several parts and the physical meaning of each of its part.

The Two Equivalent Decompositions of the Part L I

L I =L I , L +L ν I , R +L e I , R (11.2) whose terms correspond to the left doublet and the two right singlets.

The definition of L I , L is deduced from Eqs.3.6,3.5and a correspondence analog to Eq (B.15)

Frome˜1= −e 1and properties Eqs.9.5to9.10one deduces from Eqs.10.3,10.4

68 11 The Electroweak Theory in STA: Local Presentation Using again Eqs.3.6,3.5one deduces from Eqs.10.2,10.4

L e I , R =c[e μ (∂ μ ψ e )i e 3 ue˜ 0 uψ˜ e ] S (11.5b) and since e 3 u˜ = ˜u,ue˜ 0 u= ˜ue 0

L I =ce μ [(∂ μ ψ ν )i e 3 ψ˜ ν ] V +e μ [(∂ μ ψ e )i e 3 ψ˜ e ] V =L ν +L e (11.8) in conformity with Eq.10.21.

The Decomposition of the Part L II of the Lagrangian into

The following decomposition in two parts of L I I

The equation L I I = L I I , C + L I I , N (11.9) reveals an intriguing dual role of the W 3 boson It is interconnected with W 1 and W 2 through the SU(2)×U(1) gauge in the Lagrangian, while simultaneously being distinct from these two bosons, as it is featured separately in the equation's first part.

The contribution L to the lagrangian of the boson field is expressed into the sum

(for the standard presentation see [1], Eqs 7.29 and 7.40)

L I I , C and L I I , N are respectively called the charged and neutral contributions,because the boson gauge fields imply W 1 ,W 2 for L I I , C and W 3 ,B (or A and Z by

11.2 The Decomposition of the Part L I I of the Lagrangian 69

S (11.12) where[X] sc means the scalar part of X , or, recalling that i a= −ai if a∈ M,

2[(W 1 +i W 2 )j C +(W 1 −i W 2 )˜j C )] S (11.13) Introducing the “complex” (in fact real since i is real) vectorial boson gauge fields

One considers the following decomposition of L I I , N

L I I , N =L C A , N +L Z C , N (11.16) where L C A , N and L C Z , N imply the electromagnetic potential A and the Z boson gauge field, respectively.

Using g=e/sinθ,g =e/cosθand Eq.10.18one can write

(as [1] Eq 7.72) where−ej e is to be interpreted now as the current density of charge of the electron (as in [2], Eq 5.17 with e>0).

70 11 The Electroweak Theory in STA: Local Presentation and in the same way

(11.18) One can write g cosθ= e cos 2 θ sinθcosθ, g sinθ= e sin 2 θ sinθcosθ

We introduce the following “current” with the change in this equation of−j em into j e

(see [2], Eq 13.10), j N = j L 3 +2 sin 2 θj e (11.19) or (see Eq.10.12) j N = 1

This current j N ,STA form of j N C ,Eq 7.67 of [1], is equal to 2 j N C where J N C ,

Eq 13.25 of [2], is deduced from Eqs.13.1,13.6.

We obtain the contribution of the Z boson field, Eq 7.74 of [1]

The Gauges

Two gauges are present in the theory:

11.3.1 The Part U ( 1 ) of the SU ( 2 ) × ( U 1 ) Gauge

We extractce μ ã [(∂ μ ψ e )i e 3 ψ˜ e ] V from L I and−q A.j e from−L I I and we have

L U ( 1 ) =ce μ ã [(∂ μ ψ e )i e 3 ψ˜ e ] V −q A.j e , q = −e

Tiêu đề	Quantum Mechanics in the Geometry of Space–Time Elementary Theory
Tác giả	Roger Boudet
Người hướng dẫn	Roger Boudet, Honorary Professor
Trường học	Université de Provence
Thể loại	book
Năm xuất bản	2011
Thành phố	Heidelberg

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Số trang	143
Dung lượng	1,68 MB