Aspects of the standard model and quantum gravity from strand spacetime
AASPECTS OF THE STANDARD MODEL ANDQUANTUM GRAVITY FROM STRAND SPACETIME
CHARLIE BEIL
Abstract.
Strands are causal curves in spacetime with no distinct interior points,introduced to describe quantum nonlocality in a spacetime framework. We presenta model where the standard model particles are bound states of strands that inter-act by exchanging strands. In strand spacetime, it is not just the positron whoseexistence is implied by the Dirac Lagrangian: we show that hidden within thissimple Lagrangian are all the quarks, leptons, and gauge bosons, with their cor-rect spin, electric charges, color charges, and, in the electroweak sector, stability.Also encoded in the combinatorics of the Dirac Lagrangian are all the trivalentelectroweak interactions (involving both leptons and quarks), electroweak parityviolation, as well as 16 independent mass orderings that all agree exactly with ex-periment. However, the model predicts the existence of massive gluons that arecousins of the W and Z bosons, but no other particles. Using the geometry ofstrands, we are able to derive many properties of quarks, such confinement, threecolor charges, and their allowable combinations into baryons and mesons. We alsoshow that CPT invariance holds for all interactions, where C, P, and T each sitin a different connected component of the full Lorentz group. Finally, we intro-duce a quantum modification to Einstein’s equation by reinterpreting the chiraldecomposition of the Dirac Lagrangian. Contents
1. Introduction 22. Strand spacetime 33. The mass-shell condition without Lorentz violation 64. Electric and color charge from the geometry of strands 85. The chiral spinor representation of strands 126. The strand Lagrangian 147. A strand model of leptons, quarks, and gauge bosons 208. Mass orderings 249. Electroweak particle stability 2710. Electroweak parity violation 2811. A modification to Einsteins equation from the Dirac Lagrangian 3012. CPT invariance: charge conjugation as a Lorentz transformation 32
Key words and phrases.
Preon model, composite model, standard model, electroweak parity viola-tion, CPT invariance, quantum gravity, spacetime geometry. a r X i v : . [ phy s i c s . g e n - ph ] J un CHARLIE BEIL
13. The spin-statistics connection for strands 3714. Future directions: scattering with strands 38References 381.
Introduction
Strand spacetime is obtained from classical general relativity by dening the world-line of a fundamental particle to be a causal curve with no distinct interior points.Such a 1-dimensional smeared-out event in spacetime is called a strand . This ge-ometry was recently introduced to give a possible spacetime description of quantumnonlocality [B5]. In this article, we consider the following model: • Strands are ‘circular’ with spin and mass equal to their inverse radius r ,(1) m = (cid:126) cr . • Bound states of strands interact with each other by exchanging strands ac-cording to Newton’s third law of motion, called a splitting .The mass relation (1) implies that strands satisfy the modified energy-mass relation E = muc , where u is their tangential speed (Section 3). This relation in turn impliesthat the Lagrangian density for strands is(2) L = ¯ ψ ( i /∂ − m | ¯ u a u a | / ) ψ, where ¯ u and u are projections of the four-velocities of the strands represented by ¯ ψ and ψ onto a spatial hypersurface (Section 6). From this Lagrangian we obtain a newcomposite preon model where leptons, quarks, and gauge bosons are bound statesof strands, rendering the standard model Lagrangian an effective theory (Section7). The Lagrangian (2) implies that the strands themselves admit no fundamentalinteractions other than creation and annihilation in pairs. An example of scatteringusing bound states of strands is shown in Figure 1.In our model, we are able to reproduce exactly the leptons, quarks, and electroweakbosons, with their correct spin, electric charge, color charge, and, in the electroweaksector, stability (Table 4). Our model also reproduces exactly the trivalent elec-troweak Feynman interactions involving both leptons and quarks (Tables 1, 2, and6). However, our model predicts the existence of neutral and charged massive gluons,and is therefore falsiable (Table 5). It is particularly surprising that we nd much ofthe standard model – with its various particles and forces – hidden within the DiracLagrangian (2).Furthermore, the combinatorics and geometry of the model have wide explanatorypower. In particular, we obtain • the existence of electric charge and three color charges (Sections 4 and 12.2); SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 3 • the allowable combinations of quarks into baryons and mesons (Sections 4and 7.3.2); • quark connement (Sections 4 and 7.3.2); •
16 mass orderings of leptons, quarks, and gauge bosons (Section 8); • a determination of which electroweak particles are stable (Section 9); and • neutrino parity violation (Section 10).In Sections 7 - 10, we introduce the preon model and explore some of its immediateconsequences. In Section 11, we present a modification to Einstein’s equation from theDirac Lagrangian that resolves Bohr’s gedankenexperiment regarding gravitationalradiation and which-way information. In Section 12, we show that in the frameworkof strands, charge conjugation is a Lorentz transformation. Charge conjugation,parity, and time reversal are then found to each sit in a dierent connected componentof the Lorentz group O (1 , Notation:
Tensors labeled with upper and lower indices a, b, . . . represent covec-tor and vector slots using Penroses abstract index notation, and tensors labeled withindices µ, ν, . . . denote its components with respect to a coordinate basis. We use thesignature (+ , − , − , − ) throughout, and usually use natural units (cid:126) = c = 1.2. Strand spacetime
In formulating general relativity, Einstein replaced the gravitational force field inNewton’s theory of gravity with the geometry of spacetime. In a similar way, wewould like to describe the particles in the standard model not as quantized fields, but
CHARLIE BEIL time (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:93) (cid:93) (cid:28) (cid:28) (cid:63) (cid:63) (cid:127) (cid:127) ·· (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) Figure 1.
An example of photon entanglement using strands. Thecorresponding Feynman diagram is shown on the right.
Table 1.
The fundamental splittings and their O (3) particle identifications. γ Z W − ¯ ψ L ψ R ¯ ψψ = ¯ ψ L ψ R + ¯ ψ R ψ L ¯ ψ ψ := ¯ ψ L ψ R + ¯ ψ R ψ L (cid:5) (cid:5) (cid:25) (cid:25) (cid:5) (cid:5) (cid:25) (cid:25) (cid:5) (cid:5) (cid:25) (cid:25) ¯ e = ¯ ψ L ψ R = e ¯ e = ¯ ψ L ψ R = e ¯ ν e = ¯ ψ L ψ R = eν µ = ¯ ψ L ⊗ ψ L ψ R ⊗ ¯ ψ R = ¯ ν µ µ = ¯ ψ L ⊗ ψ L ψ R ⊗ ¯ ψ R = ¯ ν µ ν τ = ¯ ψ L ⊗ ¯ ψ R ψ R ⊗ ψ L = ¯ ν τ τ = ¯ ψ L ⊗ ¯ ψ R ψ R ⊗ ψ L = ¯ ν τ W + ¯ ψψ := ¯ ψ L ψ R + ¯ ψ R ψ L (cid:5) (cid:5) (cid:25) (cid:25) ¯ e = ¯ ψ L ψ R = ν e ν µ = ¯ ψ L ⊗ ψ L ψ R ⊗ ¯ ψ R = ¯ µν τ = ¯ ψ L ⊗ ¯ ψ R ψ R ⊗ ψ L = ¯ τ as geometric properties of spacetime itself. Quantum field theory would then be anemergent description of particle physics, rather than a fundamental description.To this end, we define a fundamental particle to be a causal (i.e., timelike or null)curve in spacetime that is deemed to be a single point; that is, it is a curve without SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 5
Table 2.
The non-fundamental splittings, wherein strands are allowedto be created according to the strand Lagrangian (2): in pairs of op-posite sign on a diameter. The only additional rule is that each of thefive fields ¯ ψ L/R , ψ L/R , φ is excited in some atom in the splitting. Thesplitting W ± → ZW ± is equivalent to Z → W ± W ∓ . γ Z ¯ ψ L ψ R ¯ ψψ = ¯ ψ L ψ R + ¯ ψ R ψ L (cid:5) (cid:5) (cid:25) (cid:25) (cid:5) (cid:5) (cid:25) (cid:25) µ = ¯ ψ L ⊗ ψ L ψ R ⊗ ¯ ψ R = ¯ µ µ = ¯ ψ L ⊗ ψ L ψ R ⊗ ¯ ψ R = ¯ µτ = ¯ ψ L ⊗ ¯ ψ R ψ R ⊗ ψ L = ¯ τ τ = ¯ ψ L ⊗ ¯ ψ R ψ R ⊗ ψ L = ¯ τW − = ¯ ψ ψ ¯ ψψ = W + W − = ¯ ψ ψ ¯ ψψ = W + ¯ ν e = ¯ ψ L ψ R = ν e distinct interior points. In particular, time does not flow along such a curve, even ifit is timelike; this is the source for quantum nonlocality introduced in [B5]. Definition 2.1.
Let ( ˜
M , ˜ g ) be a (3 + 1)-dimensional time-orientable Lorentzian man-ifold, which we call emergent spacetime . Let S be a collection of causal curves in ˜ M ,called strands . We declare two points x, y ∈ ˜ M to be equivalent if there is a strand α ∈ S that contains both x and y , and extend the equivalence transitively. We define spacetime to be the set of equivalence classes M := { [ x ] : x ∈ ˜ M } . Denote by π the map π : ˜ M → M, x (cid:55)→ [ x ] . Each point x in U := π ( ˜ M \ ∪ α ∈ S α ) has a unique preimage π − ( x ). Thus, to eachpoint x ∈ U , we may associate the unique vector space T x M := T π − ( x ) ˜ M .
This allows us to make the following definitions: In the framework of nonnoetherian algebraic geometry introduced in [B2], algebraic varieties withnonnoetherian coordinate rings of finite Krull dimension necessarily contain positive dimensional‘smeared-out’ points (see [B3, Theorem A] for a precise statement). Such a variety may contain,for example, curves that are identified as single points. The original purpose of this framework wasto provide a geometric description of the vacuum moduli spaces of certain unstable quiver gaugetheories in string theory [B4] (see also [B1]). It was then proposed in [B6] that this geometry couldbe applied to spacetime itself, with the hope that it could explain, in a suitable sense, quantumnonlocality.
CHARLIE BEIL • The exponential map exp : T x M → M at x ∈ U is the composition T x M = T π − ( x ) ˜ M ˜exp −→ ˜ M π −→ M. • The metric at x ∈ U is the metric at π − ( x ), g x := ˜ g π − ( x ) : T x M × T x M → R . A strand, then, is a 1-dimensional point of spacetime, and does not possess atangent space. In a given frame, a strand appears to be a classical particle, whichwe call a strand particle . The difference between a worldline and a strand is thata worldline consists of a continuum of distinct points, whereas a strand is a singlepoint.2.1.
Circular strands.
We now define the specific strands that will be used in thepreon model.
Definition 2.2.
Let β be a causal geodesic in ˜ M with affine parameterization. Astrand is circular if it is a closed segment of a curve of the form(3) α ( t ) = (cid:40) exp β ( t ) ( r cos( ωt ) e + r sin( ωt ) e ) if ˙ β ( t ) < β ( t ) ( r cos( ωt ) e + r sin( ωt ) e ) if ˙ β ( t ) = 0where r > ω ∈ R ; t is fixed; and { e , e , e } is a spacelike orthonormal basisthat is parallel transported along β , ˙ β a ∇ a e bi = 0 . For the remainder of the article, we will only consider circular strands, and so wewill usually omit the prefix ‘circular’.We define the four-momentum p a of the strand particle of α by the Planck-deBroglie relation in the direction ˙ β (with (cid:126) = c = 1): p a := k a = | ω | ˙ β a . We say the strand particle is massive if p = p a p a (cid:54) = 0, and massless if p = 0.3. The mass-shell condition without Lorentz violation
Massive strands.
Suppose α is a circular strand whose central worldline β isa timelike geodesic. In the inertial frame of β , α is a circular trajectory of radius r ,angular frequency ω , and tangential velocity u := | ˙ α | = | ω | r. We define the mass m of α to be its curvature, m := 1 r = (cid:126) cr , SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 7 with units restored in the rightmost equality. (Note that r = (cid:126) ( cm ) − is the reducedCompton wavelength of a particle of mass m .) Consequently, the rest energy E of α in the inertial frame of β is E u = (cid:126) | ω | u = (cid:126) r = mc, that is,(4) E = mcu. We thus derive a variant of Einstein’s relation E = mc ; Einstein’s relation holds ifand only if the tangential velocity u equals the speed of light c .3.0.2. Massless strands.
A circular strand particle α ( t ), with a lightlike central world-line β , cannot rotate as it propagates. Indeed, assume to the contrary that α ( t )rotates about β , just as a massive strand particle does:( α ( t ) µ ) = ( t, r cos( ωt ) , r sin( ωt ) , t ) . The tangent vector to α is then spacelike with length˙ α = − ( ωr ) . Thus, | ω | r is invariant under Lorentz transformations. But the frequency ω varieswith boosts in the z -direction, whereas the radius r is independent of such boosts.Therefore | ω | r cannot be invariant, a contradiction.The precise time at which two bound states of strands interact will always be uncer-tain (i.e., indistinguishable) over a nonzero interval of time [ t , t + (cid:15) ]. Therefore, bythe identity of indiscernibles (see [B5]), a circular strand does not actually propagateas a (0-dimensional) particle, but rather as a segment of a helix: for s ∈ [ t , t + (cid:15) ]and t ≥ t + (cid:15) , we have α ( s, t ) = (cid:40) exp β ( t ) ( r cos( ω ( t + s )) e + r sin( ω ( t + s )) e ) if ˙ β ( t ) < β ( t ) ( r cos( ω ( t + s )) e + r sin( ω ( t + s )) e ) if ˙ β ( t ) = 0However, all the interior (0-dimensional) points of the helix in ˜ M are identified asthe same point in M . As we have just shown, the helical segment rotates in aninertial frame if and only if the strand is massive. Nevertheless, a massless strandstill possesses a frequency ω , given by the pitch of its propagating helical segment.For our purposes here, it suffices to regard massive strands as simply (0-dimensional)particles.3.0.3. The mass-shell condition.
Set c = 1. From (4), the four-momentum p µ = | ω | ˙ β µ = E ˙ β µ = mu ˙ β µ of a massive strand particle satisfies p = p µ p µ = E = m u . CHARLIE BEIL
This is a modification of the standard relativistic mass-shell condition p = m .In standard quantum field theory, a particle (or field excitation) is said to be on-shell if p = m , and off-shell, or virtual, if p (cid:54) = m . During a scattering event, mostinternal particles are off-shell. Under the assumption that E = m , such particlesviolate relativity (hence the name ‘virtual’). However, under the assumption that E = mu , i.e., p = m u , off-shell massive strand particles do not violate relativity;they are simply particles whose tangential velocity is not lightlike.Consequently, a massive strand particle is- lightlike ( u = 1) iff p = m u = m ; and- timelike ( u (cid:54) = 1) iff p = m u (cid:54) = m .Furthermore, a massless strand particle is- lightlike ( u = 1) iff β is lightlike, whence p = | ω | ˙ β = 0; and- timelike ( u (cid:54) = 1) iff β is timelike, whence p = | ω | ˙ β (cid:54) = 0.The variability of u thus enables a geometric description of off-shell particles forwhich p = | ω | ˙ β always holds (that is, p = E in the massive case), and thereforerelativity is never violated .Lightlike tangential velocity may be viewed as a geodesic-like property: suppose acircle of radius r is rotating with tangential velocity u measured in an inertial frame.In the accelerated frame of the circle, Ehrenfest observed that the circumference is C = 2 πr (1 − u ) − / = 2 πrγ ( u ) . Thus, if u = 1, then C is infinite. If a circle of infinite circumference is regarded as astraight line, then the particle travels in a ‘straight line’ in its own reference frame ifand only if it travels at light speed u = 1 . Off-shell strand particles are thus unstable,and as such quickly interact with neighboring strands to recover their geodesic states.3.1.
Conservation of angular momentum.
Recall that a massive strand particle α ( t ) ∈ ˜ M has mass m = r − . Thus, the spatial angular momentum L of α ( t ), in theinertial frame of its central wordline β , equals its tangential velocity u = ωr , L = m | p | = rmu = u. Conservation of angular momentum therefore implies that the particle’s tangentialvelocity u is constant.4. Electric and color charge from the geometry of strands
In this section we will show that both electric charge and color charge, as wellas the allowable combinations into mesons and baryons, are novel features of strandspacetime geometry.The worldline α of a strand particle α ( t ) is a continuum of distinct points inemergent spacetime ˜ M , and a single point π ( α ) in spacetime M itself. Thus, there isno tangent vector field ‘along’ the point π ( α ) in M . In contrast, the strand particle SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 9 has a tangent vector field τ along its worldline α in ˜ M , since ˜ M is a manifold. Butthis four-vector is not uniquely determined by the motion of the strand particle inspacetime M , because, fundamentally, time does not flow along its worldline. Thereis therefore an ambiguity in the choice of tangent four-vector, namely τ = ± ˙ α. We identify τ = ˙ α with negative electric charge, and τ = − ˙ α with positive electriccharge. We thus obtain a new definition of electric charge from the geometry ofstrands.Consider a strand α with a timelike central worldline β . Identify the tangent spaces T β ( t ) ˜ M along β via the isomorphism induced by the tetrad { e a } , T β ( t ) ˜ M ∼ = T β ( t (cid:48) ) ˜ M ∼ = R , . Consider the spatial subspace V := span R { e , e , e } ⊂ T β ( t ) ˜ M .
Restricted to V , α has circular trajectory(5) α ( t ) = ( r cos( ωt ) , r sin( ωt ) , . For ease of notation, we assume that α has unit speed parameterization, u = ωr = 1.In isolation, or empty space, there is no distinguished direction of space. Thus,to obtain a spatial tangent vector t ∈ V to α at α ( t ), we may apply any Lorentztransformation g ∈ O (3) to the vector ˙ α ( t ) ∈ V , with the property that g is invariantunder an arbitrary Lorentz change-of-basis h ∈ O (3): h − gh = g. Consequently, g is in the center of O (3), g ∈ Z ( O (3)) = { w ± := ± diag(1 , , } ∼ = Z . The possible tangent vectors to α ( t ) are therefore t = w + ˙ α = ˙ α and t = w − ˙ α = − ˙ α . We call the choice of w + or w − the strand charge of α , denoted q ( α ), and identify thesecharges with negative and positive electric charges, respectively. We will considercharge conjugation in Section 12.2 below.Now consider a strand α in a bound state of massive strands that share a commontimelike central worldline β . Consider the worldline Frenet frame { t , n , b } of α ,translated to the origin of V . Just as there is an ambiguity in the choice of tangentvector t , namely t = ± ˙ α , there is also an ambiguity in the choice of normal vector n = ± ¨ α and binormal vector b = ± e , again since time does not flow along thetrajectory α .Indeed, the normal line L = span R { e } ⊂ V to the plane of rotation(6) P = span R { e , e } is a distinguished direction of space. Thus, to obtain the Frenet frame, we may applyany orthogonal transformation g ∈ O ( V ) = O (3) to(7) (cid:110) ˙ α , ¨ α , ˙ α × ¨ α | ˙ α × ¨ α | = sgn( ω ) e (cid:111) , with the property that g is invariant under an arbitrary orthogonal change-of-basis h in the subgroup O (2) × O (1) of O (3) specified by P , h ∈ O (2) × O (1) = O ( P ) × O ( L ) ⊂ O ( V ) . Consequently, g is in the center of O (2) × O (1), g ∈ Z ( O (2) × O (1)) ∼ = Z × Z . We denote the four elements of Z ( O (2) × O (1)), with respect to the ordered basis { e , e , e } , by(8) w ± := ± diag(1 , ,
1) and c ± := ± diag(1 , , − . These central elements act on (7) to give the Frenet frame { t , n , b } in ˜ M , and wecall the choice of central element the strand charge of α , denoted q ( α ). Thus, forexample, α has strand charge c − if and only if its Frenet frame is t = c − ˙ α = − ˙ α , n = c − ¨ α = − ¨ α , b = c − sgn( ω ) e = sgn( ω ) e . The possible Frenet frames in ˜ M are therefore(9) charge t n b w + ˙ α ¨ α sgn( ω ) e w − − ˙ α − ¨ α − sgn( ω ) e c + ˙ α ¨ α − sgn( ω ) e c − − ˙ α − ¨ α sgn( ω ) e Let P be the set of fixed planes P in a bound state ∪ α . The total charge of ∪ α isthe Z -linear combination(10) q ( ∪ α ) := (cid:88) α q ( α ) = n w w + + (cid:88) P ∈P n P c + P , where n w , n P ∈ Z are integer coefficients. A strand or bound state of strands is ableto exist in isolation if and only if it is invariant under O (3). Therefore, a bound state ∪ α may exist in isolation if and only if q ( ∪ α ) = m w w + for some m w ∈ Z . This condition restricts the allowable sets of fixed planes of abound state that is able to exist in isolation. SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 11
The simplest bound state with color charge that may exist in isolation consists oftwo strands that share the same fixed plane P , but have opposite charges c + P and c − P : c + P + c − P = 0 w + . We call such a bound state a mesonic state .The next simplest bound state with color charge that may exist in isolation consistsof three strands α , α , α , necessarily with orthogonal binormal lines, say e ( α ) = (1 , , , e ( α ) = (0 , , , e ( α ) = (0 , , . Their respective possible color charges are then r ± := ± diag( − , , g ± := ± diag(1 , − , b ± := ± diag(1 , , − r ± + g ± + b ± = w ± and(12) w + + w − = r + + r − = g + + g − = b + + b − . Therefore the strands α , α , α may have color charges q ( α ) = r + , q ( α ) = g + , q ( α ) = b + or q ( α ) = r − , q ( α ) = g − , q ( α ) = b − . We call such a bound state a baryonic state .There is a unique configuration of three pairwise orthogonal planes in R , up torotation. Thus there are precisely three orthogonal embeddings of O (2) × O (1) in O (3), up to rotation. Consequently, every O (3) bound state must be a mesonic state,a baryonic state, a collection of strands each with spatial group O (3) , or a union ofsuch states. The strand charge of a strand α is therefore an element of { w ± , r ± , g ± , b ± } . Wemake the following identifications between strand charges and electric and colorcharges: strand charge electric charge color charge w + − e (negative) w − + e (positive) r + , g + , b + red, darkgreen, blue r − , g − , b − anti-red, anti-darkgreen, anti-blue We will denote by c ± an unspecified color charge r ± , g ± , b ± .The sign of α (or q ( α )), denoted sgn( α ), is the sign ± of the superscript of q ( α ). Definition 4.1.
The antiparticle ¯ α of a strand particle α is obtained by reversingthe sign of α . The antiparticle ¯ B of a bound state B = ∪ α consisting of a collectionof strands is obtained by reversing the sign of each strand in B . Remark 4.2.
The Stueckelberg interpretation of antiparticles as particles that travelbackwards in time [St] is obtained by replacing t with − t in (5), and thus results inthe respective tangent vectors t = ˙ α and t = − ˙ α , in agreement with (9). Indeed, we have ddt ( α ( − t )) = ( − , u sin( − ωt ) , − u cos( − ωt ) ,
0) = − ddt α ( t ) . However, this interpretation does not give color charge. Furthermore, time does notflow along a strand: time does not flow backwards just as it does not flow forwards.5.
The chiral spinor representation of strands
By assumption, circular strands have spin . In this section we determine thechirality of a circular strand.Let α be a strand of radius r centered about a timelike geodesic β ( I ) ⊂ ˜ M . Identifythe tangent spaces T β ( t ) ˜ M along β via the isomorphism induced by the tetrad { e a } , T β ( t ) ˜ M ∼ = T β ( t (cid:48) ) ˜ M ∼ = R , . Consider the spatial subspace V := span R { e , e , e } ⊂ T β ( t ) ˜ M .
Restricted to V , α has circular trajectory (5). Translate the (unnormalized) worldlineFrenet frame { ˙ α , ¨ α , ˙ α × ¨ α } to the origin of V . Since time does not flow along a strand,a Frenet vector, or a wedge product of Frenet vectors, is a property of α in spacetime M if and only if it is independent of time t ; we call the set of all such wedge productsthe M -frame of α . Remark 5.1.
The parameter t is the proper time as measured by a clock alongsidethe strand particle α ( t ) in emergent spacetime ˜ M , but does not represent a timeparameter in spacetime M itself.The wordline tangent and normal vectors, ˙ α and ¨ α , vary with time t ∈ I , whereasthe binormal vector,(13) ˙ α × ¨ α | ˙ α × ¨ α | = ω r | ω r | e = sgn( ω ) e , is independent of t . Therefore, since the strand α is a single point in M , only thebinormal vector ˙ α × ¨ α of the Frenet frame belongs to the M -frame of α . SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 13
We may also consider wedge products of worldline Frenet vectors. The two wedgeproducts˙ α ∧ ( ˙ α × ¨ α ) = − ω r sin( ωt ) e ∧ e + ω r cos( ωt ) e ∧ e and ¨ α ∧ ( ˙ α × ¨ α )vary with t , whereas the two wedge products(14) ˙ α ∧ ¨ α = ω r e ∧ e and ˙ α ∧ ¨ α ∧ ( ˙ α × ¨ α ) = ω r e ∧ e ∧ e are independent of t . Thus, the two wedge products (14) also belong to the M -frameof α .To determine the possible M -frames, we act on (13) and (14) by the strand charges w ± and c ± in (8). Set(15) χ ( α ) := sgn( α ) sgn( ω ) . The electric charges w ± act byˆ b = sgn( ω ) w ± e = χ ( α ) e , ˆ t ∧ ˆ n = sgn( ω )( w ± e ) ∧ ( w ± e ) = sgn( ω ) e ∧ e , ˆ t ∧ ˆ n ∧ ˆ b = sgn( α ) e ∧ e ∧ e . (16)Similarly, the color charges c ∓ act byˆ b = sgn( ω ) c ∓ e = χ ( α ) e , ˆ t ∧ ˆ n = sgn( ω )( c ∓ e ) ∧ ( c ∓ e ) = sgn( ω ) e ∧ e , ˆ t ∧ ˆ n ∧ ˆ b = sgn( α ) e ∧ e ∧ e . (17)There are therefore four possible M -frames, and these are specified by the signs of α and ω : { ˆ b , ˆ t ∧ ˆ n , ˆ t ∧ ˆ n ∧ ˆ b } = { χ ( α ) e , sgn( ω ) e ∧ e , sgn( α ) e ∧ e ∧ e } . We now briefly recall the chiral decomposition of the Lorentz algebra. The gener-ators of the Lorentz algebra so (1 , J i and three boosts K i , admit linear combinations A i := ( J i + iK i ) and B i := ( J i − iK i ) that satisfy[ A i , A j ] = i(cid:15) ijk A k , [ B i , B j ] = i(cid:15) ijk B k , [ A i , B j ] = 0 . Therefore the complexification so (1 , C := so (1 , ⊗ R C decomposes as a direct sum(18) so (1 , C ∼ = su (2) C ⊕ su (2) C . Since there is a bijection between real representations of a real Lie algebra and com-plex representations of its complexification, (18) implies that so (1 ,
3) and su (2) ⊕ su (2) The generators satisfy[ K i , K j ] = − i(cid:15) ijk J k , [ J i , J j ] = i(cid:15) ijk J k , [ J i , K j ] = i(cid:15) ijk K k . have the same irreducible representations. The chiral spinors ψ L and ψ R live in therepresentations of the left and right summands of su (2) ⊕ su (2) respectively.Again consider the strand α . We may represent α by a left or right chiral spinor ψ L := P L ψ = (1 − γ ) ψ or ψ R := P R ψ = (1 + γ ) ψ, where ψ is a (4-component) Dirac spinor. The chirality of α is obtained by determin-ing which su (2) C subgroup of the complexified Lorentz algebra so (1 , C its associatedlinear combination C α = sgn( ω ) J z + i sgn( α ) K z belongs. We have C α ∈ su (2) C ⊕ (cid:0) , (cid:1) ⇐⇒ sgn( α ) sgn( ω ) = +1 ,C α ∈ ⊕ su (2) C = (cid:0) , (cid:1) ⇐⇒ sgn( α ) sgn( ω ) = − . Whence, by (15), the chiral spinor representing α is (cid:40) ψ L if χ ( α ) = +1 ψ R if χ ( α ) = − α is therefore given by χ ( α ).By (16) and (17), the sign of α is the sign of the top wedge product ˆ t ∧ ˆ n ∧ ˆ b . Itfollows that sgn( α ) is also the handedness, left or right, of the ordered basis { t , n , b } . We identify the phase of the spinor ψ L,R with the strand particle α ( t ) ∈ ˜ M itself,by the linear isomorphism from the plane of rotation P in (6) to C , e (cid:55)→ , e (cid:55)→ i. Under this isomorphism, β ( t ) is the origin of C .6. The strand Lagrangian
In Section 2, we defined the mass of a strand α to be equal to its inverse radius m := r − . In Section 3, we showed that this definition implies that α has energy E = mu , where u is it tangential speed, rather than E = m . Based on this relation,we model strand interactions by the Lagrangian density(19) L ( β ( t )) := ¯ ψ ( i /∂ − m | ¯ u a u a | / ) ψ, where ¯ ψ , ψ are Dirac spinors that represent strand particles ¯ α , α on a circle withcentral worldline β ; and ¯ u a , u a are the projections in T β ( t ) ˜ M of the four-velocities(based at the origin of T β ( t ) ˜ M ) of ¯ α , α onto the spatial hypersurface of the inertialframe of β .Note that the mass dimensions of the fields are [ ψ ] = and [ u a ] = 0, and thus[ m ] = 1 as it should be. SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 15
In Section 4 we found that strands are represented by chiral spinors, ψ L = P L ψ or ψ R = P R ψ . Expanding the Lagrangian (19), we have L (i) = i ¯ ψ L /∂ψ L − m | ¯ u a u a | / ¯ ψ L ψ R + ( L ↔ R ) (ii) = i ¯ ψ L /∂ψ L − mφ ¯ ψ L ψ R + ( L ↔ R ) , (20)where ( i ) holds since¯ ψ L := ( ψ L ) † γ = ¯ ψP R and ¯ ψ R = ¯ ψP L . In ( ii ), φ := | ¯ u a u a | / is viewed as a real scalar field. Each chiral spinor ¯ ψ L , ψ R ,¯ ψ R , ψ L represents a strand ¯ α L , α R , ¯ α R , α L of radius r and angular velocity ¯ ω L , ω R ,¯ ω R , ω L . Since the terms ¯ ψ L ψ R and ¯ ψ R ψ L have the same coupling constant mφ , therespective spatial four-velocities ¯ u a , u a , ¯ v a , v a satisfy | ¯ u a u a | / = φ = | ¯ v b v b | / . Symmetric atoms.Definition 6.1.
We call a bound state of strands that share a common plane ofrotation P and central worldline β an atom . An atom is symmetric if it is an excitationof the mass terms mφ ¯ ψψ = mφ ( ¯ ψ L ψ R + ¯ ψ R ψ L ). We call the scalars ¯ ψ L ψ R and ¯ ψ R ψ L the diameters of a symmetric atom.The symmetric atoms where each strand has O (3) charge are given by the following:¯ ω L/R = ω R/L ¯ u a u a = 1: on-shell γ or Z boson0 < | ¯ u a u a | <
1: off-shell γ or Z boson¯ u a u a = 0: ¯ α L/R , α R/L are free: a splitting occurs¯ u a u a = −
1: an apex : the creation/annihilat. of ¯ α L/R , α R/L ¯ ω L/R = − ω R/L ¯ u a u a = − r cos θ : W + or W − boson, on-shell if and only if r = 1¯ u a u a = 0: ¯ α L/R , α R/L are free: a splitting is possible • First suppose ¯ ω L/R = ω R/L and ¯ u a u a = 1. Then φ = 1 does not vary.(i) There are two such atoms consisting of two strands, differing in their directionof rotation: γ = ¯ ψ L ψ R : (cid:35) (cid:32) (cid:79) (cid:79) (cid:15) (cid:15) ¯ ψ L ψ R (cid:32) (cid:35) (cid:15) (cid:15) (cid:79) (cid:79) ¯ ψ L ψ R Since the strands ¯ α L , α R have opposite chirality and ¯ ω L = ω R , the two strands haveopposite sign by the chirality relation (15):sgn( ¯ α L ) sgn(¯ ω L ) = χ ( ¯ α L ) = − χ ( α R ) = − sgn( α R ) sgn( ω R ) . Thus, the strands have opposite w ± charges, and so their bound state ¯ ψ L ψ R haszero charge by (10). We identify these two atoms with photons of opposite circularpolarization (for non-circular polarization, see [B5]).(ii) There is also an atom with ¯ ω L/R = ω R/L consisting of four strands: Z = ¯ ψψ = ¯ ψ L ψ R + ¯ ψ R ψ L : (cid:35) (cid:79) (cid:79) (cid:32) (cid:15) (cid:15) (cid:32) (cid:111) (cid:111) (cid:35) (cid:47) (cid:47) ¯ ψ L ψ R ¯ ψ R ψ L This atom also has total charge zero, and so we identify it with the Z boson. The Z atom is not a superposition of two photons because the two diameters are boundtogether. In particular, there is binding energy that contributes to the atom’s mass;see Section 8. • Now suppose ¯ ω L/R = − ω R/L . Then φ = | cos θ | , and in particular the couplingconstant φm varies in time. From the associativity ( φm ) ¯ ψψ = φ ( m ¯ ψψ ), there aretwo possible representations of such atoms:- A coupling representation , where the atom is represented by ( φm ) ¯ ψψ . Thisrepresentation allows to determine certain mass orderings (Section 8), andprovides an explicit derivation of electroweak parity violation (Section 10).- A field representation , where the atom is represented by φ ( m ¯ ψψ ), with φ anindependent real scalar field bound to a strand in the γ or Z atom m ¯ ψψ .Since φ varies in time, φ has a time orientation. Thus φ has a sign,sgn( φ ) = ± , as described in Section 4. This representation therefore shows that thereshould be an additional O (3) charge – the charge of the scalar field φ – in anatom with a time varying coupling constant.Both representations are useful, as they reveal different aspects of such atoms.The possible atoms with ¯ ω L/R = − ω R/L are given in Table 3. We assume that thescalar field φ only binds to the strand of opposite sign in the photon diameter ¯ ψ L ψ R of the Z atom, since the strands in the non-photon diameter ¯ ψ R ψ L may mutuallyannihilate in a splitting (see Definition 7.1 below).(i) First consider the atom B = φm ¯ ψ L ψ R . In the coupling representation B = ( φm ) ¯ ψ L ψ R ,B does not have circular polarization since ¯ ω L = − ω R . In contrast, in the fieldrepresentation B = φ ( m ¯ ψ L ψ R ) ,B is a bound state of the scalar field φ with the photon atom γ = ¯ ψ L ψ R , and thus B has circular polarization. But this implies that B is inconsistent (that is, ill-defined) SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 17
Table 3.
The possible O (3) symmetric atoms with a time varyingcoupling constant φm .coupling representation field representationinconsistent : (cid:32) (cid:79) (cid:79) (cid:32) (cid:79) (cid:79) ¯ ψ L ψ R (cid:54) = (cid:71)(cid:35) (cid:79) (cid:79) (cid:32) (cid:15) (cid:15) ¯ ψ L = φ ¯ ψ L ψ R ( φm ) ¯ ψ L ψ R = φ ( m ¯ ψ L ψ R ) = m ¯ ψ L ψ R W − : (cid:32) (cid:79) (cid:79) (cid:32) (cid:79) (cid:79) (cid:35) (cid:111) (cid:111) (cid:35) (cid:111) (cid:111) ¯ ψ L ψ R ¯ ψ R ψ L = (cid:71)(cid:35) (cid:79) (cid:79) (cid:32) (cid:15) (cid:15) (cid:32) (cid:111) (cid:111) (cid:35) (cid:47) (cid:47) ¯ ψ L = φ ¯ ψ L ψ R ¯ ψ R ψ L ( φm ) ¯ ψψ = φ ( m ¯ ψψ ) = m ( ¯ ψ L ψ R + ¯ ψ R ψ L ) W + : (cid:32) (cid:79) (cid:79) (cid:32) (cid:79) (cid:79) (cid:35) (cid:111) (cid:111) (cid:35) (cid:111) (cid:111) ¯ ψ L ψ R ¯ ψ R ψ L = (cid:35) (cid:79) (cid:79) (cid:71)(cid:35) (cid:15) (cid:15) (cid:32) (cid:111) (cid:111) (cid:35) (cid:47) (cid:47) ¯ ψ L ψ R = φψ R ¯ ψ R ψ L ( φm ) ¯ ψψ = φ ( m ¯ ψψ ) = m ( ¯ ψ L ψ R + ¯ ψ R ψ L )since ( φm ) ¯ ψ L ψ R = φ ( m ¯ ψ L ψ R ). We therefore conclude that this atom cannot bephysical.(ii) Now consider the atom B = φm ¯ ψψ . Since the strands ¯ α L , α R have oppositechirality and ¯ ω L = − ω R , the chirality relation (15) implies that sgn( ¯ α L ) = sgn( α R ).Similarly sgn( ¯ α R ) = sgn( α L ). Thus each diameter has charge 2 w + or 2 w − . Therefore,a priori, B has charge q ( B ) = 4 w ± + w sgn( φ ) or q ( B ) = 0 + w sgn( φ ) , by (10). However, in the field representation B = φ ( m ¯ ψψ ), we find that the atomis a bound state of a Z = ¯ ψψ atom, which has zero charge, and the scalar field φ ,which has charge w + or w − . Therefore q ( B ) = q ( Z ) + q ( φ ) = w sgn( φ ) . Thus, the two diameters in the coupling representation B = ( φm ) ¯ ψψ must haveopposite charge, 2 w + and 2 w − . Consequently, the charge of the atom arises entirelyfrom the particular periodic configuration of the strands. We identify these twosymmetric atoms with the W − and W + particles. time (cid:79) (cid:79) (cid:32)(cid:32) (cid:32)(cid:32) (cid:32)(cid:32) (cid:32)(cid:32) (cid:32) mm mm mm mm m ψ L (cid:105) (cid:105) ψ R (cid:53) (cid:53) ψ L (cid:105) (cid:105) ψ R (cid:53) (cid:53) ψ L (cid:105) (cid:105) ψ R (cid:53) (cid:53) ψ L (cid:105) (cid:105) ψ R (cid:53) (cid:53) (cid:32) (cid:32) m m ψ L (cid:72) (cid:72) ψ R (cid:35) (cid:35) ψ L (cid:81) (cid:81) (i) (ii) Figure 2. (i) The standard interpretation of the chiral decompositionof the Dirac Lagrangian: what Penrose refers to as a ‘zig-zag path’. (ii)The interpretation of the chiral decomposition in the context of strands:strands of opposite sign can be created or annihilated at apexes.6.2.
The creation and annihilation of strands: apexes.
Suppose ¯ ω L/R = ω R/L and ¯ u a u a = −
1. This configuration describe two strand particles ¯ α , α on a diame-ter whose spatial tangent vectors ¯ u a , u a are parallel. Furthermore, by the chiralityrelation (15), ¯ α and α have opposite sign. The configuration therefore describes thecollision of two strands of opposite sign: (cid:35)(cid:32) (cid:79) (cid:79) ¯ ψ L ψ R Definition 6.2. An apex is a point x ∈ ˜ M where two strands, or two coupling fields,of opposite sign on a diameter are created or annihilated; see Figure 2.Two strands that meet at an apex and belong to two atoms in a splitting maybe transformed into a single strand by reversing the time orientation of one of theatoms; see Remark 7.2.Energy-momentum is conserved in interactions between bound states of strands.Thus, if an apex appears in such an interaction, then the newly created strands willobtain their energy from other strands involved in the interaction. In particular,unlike particle-antiparticle creation, the strands will not use ‘free energy’ from thevacuum, allowed by the time-energy uncertainty principle, to exist. Therefore, thetime-energy uncertainty principle does not constrain the lifetime of strands.6.3. A remark on the quantization of the Lagrangian.
From Dirac’s path in-tegral formulation of quantum theory, Feynman concluded that ‘Nature takes every
SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 19
Table 4.
Particle identifications of the symmetric and split atoms a t o m r o t a t i o n a l s y mm e t r y ⇒ s p i n > li k e c h a r g e s ⇒ s t a b ili t y s t r a nd c h a r g e ⇒ e l ec t r i cc h a r g e p a r t i c l e (cid:35) (cid:32) (cid:79) (cid:79) (cid:15) (cid:15) ¯ ψ L ψ R π no stable - 0 c +1 + c − γ ˜ γ (cid:35) (cid:79) (cid:79) (cid:32) (cid:15) (cid:15) (cid:32) (cid:111) (cid:111) (cid:35) (cid:47) (cid:47) ¯ ψψ = ¯ ψ L ψ R + ¯ ψ R ψ L π yes unstable - 0 c +1 + c − Z ˜ Z (cid:32) (cid:79) (cid:79) (cid:32) (cid:79) (cid:79) (cid:35) (cid:111) (cid:111) (cid:35) (cid:111) (cid:111) = (cid:71)(cid:35) (cid:79) (cid:79) (cid:32) (cid:15) (cid:15) (cid:32) (cid:111) (cid:111) (cid:35) (cid:47) (cid:47) ( φm ) ¯ ψψ = φ ( m ¯ ψψ )= ¯ ψ L ψ R + ¯ ψ R ψ L π yes unstable - w + c +1 + c − + w + − − W − ˜ W − (cid:32) (cid:15) (cid:15) ψ R π no stable - w + c + − − ed (cid:71)(cid:35) (cid:79) (cid:79) (cid:32) (cid:111) (cid:111) ¯ ψ L ⊗ ψ L π yes unstable - w + c + − − µs (cid:71)(cid:35) (cid:79) (cid:79) (cid:32) (cid:47) (cid:47) ¯ ψ L ⊗ ¯ ψ R π yes unstable - w + c + − − τb (cid:71)(cid:35) (cid:15) (cid:15) ψ R π no stable - 0 c + + w − ν e u (cid:35) (cid:79) (cid:79) (cid:32) (cid:111) (cid:111) ¯ ψ L ⊗ ψ L π no stable - 0 c + + w − ν µ c (cid:35) (cid:79) (cid:79) (cid:32) (cid:47) (cid:47) ¯ ψ L ⊗ ¯ ψ R π no stable - 0 c + + w − ν τ t possible path’. However, in [B5] we introduced the possibility that the geodesic hy-pothesis of general relativity is also able to describe quantum phenomena, by propos-ing the following alternative: Nature takes a set of all indistinguishable stationary paths.Thus, by the identity of indiscernibles, Nature takes a single stationary path.
Classical physics and quantum physics would then share the same underlying prin-ciple. Furthermore, path superposition would result whenever two paths, that areindistinguishable to all the constituents of the universe, become distinguishable at acollective, emergent scale.In this framework, it may be that the strand Lagrangian L should not be quantized.This would be possible if the classical equations of motion of L correspond to some‘quantum motion’ determined by the path integral of a different Lagrangian. Indeed,the equations of motion of the Lagrangian L = ¯ ψ ( i /∂ − mu ) ψ, namely i /∂ψ = muψ (that is, E = mu ), correspond to certain quantum perturbationsof the equations of motion of the Dirac Lagrangian, L Dirac = ¯ ψ ( i /∂ − m ) ψ, namely i /∂ψ (cid:54) = mψ , whenever u (cid:54) = 1.One stark difference that would remain, however, is that there are vacuum fluc-tuations from the path integral, but not from the Lagrangian alone. In our model,then, apexes would only arise so that strands that are off-shell can become on-shell,and would not spontaneously occur without cause. We leave these speculations for future work.7.
A strand model of leptons, quarks, and gauge bosons
A primary objective of our model is to provide a spacetime description of quantumnonlocality, and this would not be possible if each elementary particle is simply adifferent type of strand; see [B5]. We therefore introduce a new preon model ofparticle physics, where leptons, quarks, and gauge bosons are bound states of strandsthat interact by exchanging strands.7.1.
Split atoms.
We make the following assumption:(21)
Strands of opposite sign attract, and strands of the same sign repel.
In the standard model, the physical mechanism that causes the attraction and repul-sion between electric charges is the exchange of photons. This will also be the causeof attraction and repulsion between charged strand atoms in our model. However, Vacuum fluctuations account for the Casimir force and the positive cosmological constant, aswell as provide a physical mechanism that corrects bare masses to renormalized masses. Furtherdevelopment of our framework is therefore required to address these issues.
SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 21 we do not know what physical mechanism may cause the attraction and repulsionbetween the individual strands within a single atom (though we expect that it maybe related to the microscopic curvature of spacetime). Our model therefore requiresfurther development to address this question.
Definition 7.1.
A symmetric atom may undergo a fundamental splitting into two split atoms if φ = 0, that is, if the strands in each diameter are no longer boundtogether, such that the following holds.(i) The strands in the non-photon diameter ¯ ψ R ψ L of the symmetric atom mayannihilate each other at an apex.(ii) Newton’s third law of motion:
One strand from each non-annihilated diameterof the symmetric atom belongs to each split atom.(iii) By (21), the two strands in a split atom have opposite sign.A non-fundamental splitting occurs if (i) - (iii) hold, and new strands are createdat apexes. We impose one additional rule for such splittings:(iv) Each of the five fields ¯ ψ L/R , ψ L/R , φ is excited in some atom in the splitting.The set of all fundamental and non-fundamental splittings are shown in Tables 1 and2 respectively. Remark 7.2.
An incoming (outgoing) atom in a splitting can be transformed intoan outgoing (resp. incoming) atom by reversing the time orientation, that is, the sign,of each strand in the atom. In particular, if B → B B is a splitting of a symmetricatom B , then ¯ B B → B and ¯ B → ¯ B B are also allowed interactions (just as isthe case for Feynman interactions).The vertices in a Feynman diagram represent splittings. Thus, the total four-momentum is conserved at each splitting, (cid:88) incoming p µ = (cid:88) outgoing p (cid:48) µ . However, the four-momentum along a single strand in a splitting need not be con-served; momentum of one strand may be transferred to another strand during asplitting.With these rules, we reproduce exactly the leptons, quarks, and electroweak bosons,with their correct spin, electric and color charges, certain mass orderings, and, inthe electroweak sector, stability, as well as the electroweak interactions. Note thatrule (iv) ensures that there are no photon self-interactions, and no photon-neutrinointeractions.7.2.
Spin.
Recall that a circular strand has spin , by assumption. The spin of adiameter ¯ ψ L/R ψ R/L is zero since it is a scalar field. However, the input to the couplingconstant φm requires the specification of the plane of rotation P , which is determined by the spatial four-vector e a . Thus, although a diameter is a scalar field, symmetricatoms have spin 1.Split atoms also have a plane of rotation P , and so are not scalar fields. In par-ticular, split atoms do not have spin 0. But a split atom can bind with another splitatom to form a symmetric atom, which has spin 1. Therefore split atoms must havespin .In the defining representation of SO (3), a vector is returned to its initial positionby a rotation of θ = 2 π , whereas in the spin- representation of SO (3), a spinoris returned to its initial position by a rotation of 4 π = 2 θ . The ratio of rotationalsymmetry between vectors and spinors, namely 2, is precisely the ratio of rotationalsymmetry between symmetric and split atoms; see Table 1.7.3. Particle identifications.
Based on each atom’s spin, electric charge, and colorcharge, we make the particle identifications given in Table 4.
Remark 7.3.
In our model, photons do not a priori interact with all electricallycharged atoms; instead, they interact only with those atoms that together satisfy thesplitting rules. From these rules, we find that a photon is able to interact with alepton atom if and only if it has a nonzero electric charge, but this is not the case forquark atoms.7.3.1.
Lepton interactions.
There are four O (3) symmetric atoms: the photon γ , Z -boson, and W ± -bosons, shown in Table 4. Their splittings into split atoms are shownin Tables 1 and 2. The splittings correspond precisely to the Feynman interactionsbetween leptons and electroweak gauge bosons.We leave the question of neutrino oscillation (and quark oscillation) for futurework.7.3.2. Quark interactions.
The photon diameter ¯ ψ L ψ R specifies an atom’s plane ofrotation P , since the strands in the photon diameter cannot mutually annihilate.Thus the spatial group of the strands in the photon diameter, O (2) or O (3), isdetermined by whether P is fixed in a bound state with other atoms (forming ameson or baryon), or whether P is unconstrained. However, the spatial group of thestrands in the non-photon diameter ¯ ψ R ψ L is always O (3) because P is fixed entirelyby the photon diameter. Therefore the strands in the photon diameter, namely ¯ α L and α R , can carry either O (2) (color) charge or O (3) (electric) charge, whereas thestrands in the non-photon diameter, ¯ α R and α L , can only carry O (3) charge. Definition 7.4.
If the strands in the photon diameter ¯ ψ L ψ R have O (2) charge, thenwe say the atom is an O (2) atom; otherwise it is an O (3) atom.We call the O (2) symmetric atoms gluons , though they differ from gluons in quan-tum chromodynamics. There are three types of gluons: This is not a rigorous argument, but in order to reproduce the correct quark charges, we want onlythe strands in the photon diameter to be able to carry O (2) charge. SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 23 • Neutral massless gluons ˜ γ , called γ -gluons , which mediate color charge be-tween d , s , and b quarks. • Neutral massive gluons ˜ Z , called Z -gluons , which mediate color charge be-tween all quarks. • Charged massive gluons ˜ W , called W -gluons , which allow flavor transforma-tions within a generation.The model therefore predicts the existence of both neutral and charged massive glu-ons. Their splittings are shown in Tables 5. Note that a γ -gluon (resp. Z -gluon, W -gluon) with strands of equal color is simply a photon (resp. Z , W boson), by therelations (12); see Table 6.In our model, quarks do not have fractional electric charge as they do in QCD.Instead, they possess integer combinations of strand charges. Nevertheless, our modelgives the correct electric charges for all baryons and mesons: Proposition 7.5.
Upon substituting the charges w ± (cid:55)→ ∓ and c ± (cid:55)→ ∓ in Table 4, we obtain the fractional electric charges of the quarks in QCD. Therefore,the strand model and QCD predict the same electric charges for all baryons andmesons.Proof. The second statement follows from the relations (11) and (12). (cid:3)
Remark 7.6.
Our model ‘explains’ two of the prominent features of QCD: • The reason there are three color charges is because there are three dimensionsof space: specifically, there are three pairwise orthogonal embeddings of O (2) × O (1) into O (3). • The reason that quarks cannot exist in isolation is because color charge is onlypossible when there is a distinguished direction of space; in isolation, there isno distinguished direction of space.However, the different gluon types predicted by the model do not arise in QCD, andthis feature may cause the model to fail.7.3.3.
Gauge boson interactions.
Our model reproduces exactly the standard modeltrivalent vertices γW + W − and ZW + W − involving the electroweak gauge bosons;these are shown in Table 2. The four-valent electroweak vertices γγW + W − , γZW + W − , ZZW + W − , W + W − W + W − , are obtained by a (simultaneous) composition of two of the splittings { γ → W + W − } ∼ = { W ± → γW ± } or { Z → W + W − } ∼ = { W ± → ZW ± } . Furthermore, our model predicts the new photon-gluon vertices, shown in Table 6: γ ˜ W + ˜ W − , Z ˜ W + ˜ W − , W ± ˜ Z ˜ W ± . Table 5.
The splittings of the O (2) symmetric atoms (gluons) into O (2) atoms. The strands that do not belong to the original symmetricatom are paired by opposite sign on a diameter. ˜ γ ˜ Z ˜ W − ¯ ψ cL ψ c (cid:48) R ¯ ψ c ψ c (cid:48) := ¯ ψ cL ψ c (cid:48) R + ¯ ψ R ψ L ¯ ψ c ψ c (cid:48) := ¯ ψ cL ψ c (cid:48) R + ¯ ψ R ψ L (cid:5) (cid:5) (cid:25) (cid:25) (cid:5) (cid:5) (cid:25) (cid:25) (cid:5) (cid:5) (cid:25) (cid:25) ¯ d = ¯ ψ cL ψ c (cid:48) R = d ¯ d = ¯ ψ cL ψ c (cid:48) R = d ¯ u = ¯ ψ cL ψ c (cid:48) R = dc = ¯ ψ cL ⊗ ψ L ψ c (cid:48) R ⊗ ¯ ψ R = ¯ c s = ¯ ψ cL ⊗ ψ L ψ c (cid:48) R ⊗ ¯ ψ R = ¯ ct = ¯ ψ cL ⊗ ¯ ψ R ψ c (cid:48) R ⊗ ψ L = ¯ t b = ¯ ψ cL ⊗ ¯ ψ R ψ c (cid:48) R ⊗ ψ L = ¯ ts = ¯ ψ cL ⊗ ψ L ψ c (cid:48) R ⊗ ¯ ψ R = ¯ s s = ¯ ψ cL ⊗ ψ L ψ c (cid:48) R ⊗ ¯ ψ R = ¯ s W + b = ¯ ψ cL ⊗ ¯ ψ R ψ c (cid:48) R ⊗ ψ L = ¯ b b = ¯ ψ cL ⊗ ¯ ψ R ψ c (cid:48) R ⊗ ψ L = ¯ b ¯ ψ c ψ c (cid:48) := ¯ ψ cL ψ c (cid:48) R + ¯ ψ R ψ L ˜ W − = ¯ ψ c ψ d ¯ ψ d ψ c (cid:48) = ˜ W + ˜ W − = ¯ ψ c ψ d ¯ ψ d ψ c (cid:48) = ˜ W + (cid:5) (cid:5) (cid:25) (cid:25) ¯ u = ¯ ψ cL ψ c (cid:48) R = u ¯ d = ¯ ψ cL ψ c (cid:48) R = uc = ¯ ψ cL ⊗ ψ L ψ c (cid:48) R ⊗ ¯ ψ R = ¯ st = ¯ ψ cL ⊗ ¯ ψ R ψ c (cid:48) R ⊗ ψ L = ¯ b Mass orderings
In this section, we show that our model produces 16 independent mass orderingsthat agree exactly with the standard model particle masses. We consider three sourcesof mass in a strand atom: • angular momentum; • the coupling field φ ; and • binding energy: the number of pairs of strands that do not belong to the samediameter.In the following, we use these sources, together with Table 4, to derive certain massorderings of leptons, quarks, and gauge bosons.8.1. Angular momentum.
Consider two atoms B , C that are identical except thatthe diameters of B rotate in opposite directions, clockwise and counter-clockwise,whereas the diameters of C rotate in the same direction. Then the total angularmomentum of B will be zero, and that of C will be nonzero, L ( B ) = 0 and L ( C ) > , where L = | L | is the magnitude of the (spatial) angular momentum L . Furthermore,the angular momentum L contributes to the rest energy of the atom. Therefore, themass of B must be less than the mass of C , m ( B ) < m ( C ). We find that this agreeswith the experimentally determined mass values: SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 25
Table 6.
The splittings of the O (3) symmetric atoms into O (2) atoms.These splittings all use the relation w + + w − = c + + c − in (12). Thestrands that do not belong to the original symmetric atom are paired byopposite sign on a diameter. Note that the photon γ does not interactwith the quarks u , c , t , in contrast to the standard model. γ Z W − ¯ ψ L ψ R ¯ ψψ = ¯ ψ L ψ R + ¯ ψ R ψ L ¯ ψ ψ := ¯ ψ L ψ R + ¯ ψ R ψ L = = = ¯ ψ cL ψ cR ¯ ψ c ψ c := ¯ ψ cL ψ cR + ¯ ψ R ψ L ¯ ψ c ψ c := ¯ ψ cL ψ cR + ¯ ψ R ψ L (cid:5) (cid:5) (cid:25) (cid:25) (cid:5) (cid:5) (cid:25) (cid:25) (cid:5) (cid:5) (cid:25) (cid:25) ¯ d = ¯ ψ cL ψ cR = d ¯ d = ¯ ψ cL ψ cR = d ¯ u = ¯ ψ cL ψ cR = dc = ¯ ψ cL ⊗ ψ L ψ cR ⊗ ¯ ψ R = ¯ c s = ¯ ψ cL ⊗ ψ L ψ cR ⊗ ¯ ψ R = ¯ ct = ¯ ψ cL ⊗ ¯ ψ R ψ cR ⊗ ψ L = ¯ t b = ¯ ψ cL ⊗ ¯ ψ R ψ cR ⊗ ψ L = ¯ ts = ¯ ψ cL ⊗ ψ L ψ cR ⊗ ¯ ψ R = ¯ s s = ¯ ψ cL ⊗ ψ L ψ cR ⊗ ¯ ψ R = ¯ s W + b = ¯ ψ cL ⊗ ¯ ψ R ψ cR ⊗ ψ L = ¯ b b = ¯ ψ cL ⊗ ¯ ψ R ψ cR ⊗ ψ L = ¯ b ¯ ψ c ψ c := ¯ ψ cL ψ cR + ¯ ψ R ψ L ˜ W − = ¯ ψ c ψ d ¯ ψ d ψ c (cid:48) = ˜ W + ˜ W − = ¯ ψ c ψ d ¯ ψ d ψ c (cid:48) = ˜ W + (cid:5) (cid:5) (cid:25) (cid:25) ¯ u = ¯ ψ cL ψ cR = u ¯ d = ¯ ψ cL ψ cR = uc = ¯ ψ cL ⊗ ψ L ψ cR ⊗ ¯ ψ R = ¯ st = ¯ ψ cL ⊗ ¯ ψ R ψ cR ⊗ ψ L = b L = 0 L > m ( ν µ ) < m ( ν τ ) m ( c ) < m ( t ) m ( µ ) < m ( τ ) m ( s ) < m ( b )Two of these inequalities, for example m ( µ ) < m ( τ ) and m ( s ) < m ( b ), are notretrodictions because we could swap the atoms labeled µ and τ , and the atomslabeled s and b . The other two inequalities, however, are fixed by the W ± splittings,and so m ( c ) < m ( t ) is an honest retrodiction, and m ( ν µ ) < m ( ν τ ) is a prediction. Furthermore, the individual diameters of the Z atom each have nonzero angularmomentum, whereas the diameters of the W atom have zero angular momentum. We note that, in the standard model, neutrino flavor eigenstates and mass eigenstates do notcoincide in order to account for neutrino oscillations, which differs from our model.
Therefore we should have(22) m ( W ) < m ( Z ) , which also agrees with experiment.8.2. The coupling field φ . In the field representation, the only difference betweenthe W atom and the Z atom is that the W atom contains the scalar coupling field φ = | u a u a | . However, the mass of the W atom is less than the mass of the Z atom,by (22). Therefore the field φ must have negative mass, m ( φ ) <
0. This presents noproblem, however, because φ cannot exist in isolation, as it is an emergent propertyof atoms for which ¯ ω L/R = − ω R/L .Since the field φ has negative mass, in general adding φ to an atom which does notcontain φ will decrease the atom’s mass .However, suppose both strands in a split atom have O (3) charge; then one strandwill have charge w + and the other will have charge w − by (21). Since φ also hascharge w + or w − , and ‘like charges repel’, a minimum energy E φ will be required tokeep φ in the atom. In our model, we assume that this energy is greater than theabsolute value of the mass of φ , E φ > | m ( φ ) | . Thus, if both strands have nonzero O (3) charge, then adding the coupling field tothe atom will increase its mass, rather than decrease it.We find exact agreement with experiment:At most one diameter with O (3) charge: Both diameters with O (3) charge:with φ without φ without φ with φm ( ν e ) < m ( e ) m ( ν µ ) < m ( µ ) m ( u ) < m ( d ) m ( ν τ ) < m ( τ ) m ( s ) < m ( c ) m ( b ) < m ( t )8.3. Binding energy.
Let B be a strand atom, and let n ( B ) be the number of pairsof strands of B that do not belong to the same diameter. Each such pair requiresenergy to bind the strands together in the atom. Thus, if two atoms B , C satisfy n ( B ) < n ( C ) , then C should require more binding energy than B . Therefore, if B and C are ‘sufficiently similar’ atoms, then n ( B ) < n ( C ) will imply m ( B ) < m ( C ).We find exact agreement with experiment if we consider all O (3) (resp. O (2)) atomscontaining precisely two charges: SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 27 n ( B ) < n ( C ) m ( γ ) < min { m ( ν µ ) , m ( ν τ ) } m (˜ γ ) < min { m ( c ) , m ( t ) } ∗ m ( ν e ) < min { m ( ν µ ) , m ( ν τ ) } m ( u ) < min { m ( c ) , m ( t ) } We also find exact agreement with experiment if we consider all O (3) (resp. O (2))atoms containing the coupling field φ : n ( B ) < n ( C ) m ( ν e ) < min { m ( µ ) , m ( τ ) } m ( u ) < min { m ( s ) , m ( b ) } max { m ( µ ) , m ( τ ) } < m ( W )max { m ( s ) , m ( b ) } < m ( ˜ W ) ∗ The two inequalities marked ( ∗ ) are predictions for the ˜ γ and ˜ W gluons, as theseparticles do not belong to the standard model.9. Electroweak particle stability
First consider the electroweak sector. By (21), an O (3) atom is unstable if andonly if it contains more than two like charges w ± . Using Table 4, we findstable particles: electron, electron neutrino, muon neutrino, tau neutrino,photonunstable particles: muon, tau, W boson, Z bosonThis classification of stability is in exact agreement with experiment.Now consider the quark sector. The atom consisting of a single strand is an (anti-)electron if it has spatial group O (3), and an (anti-)down quark if it has spatial group O (2). According to the strand Lagrangian (19), both types of strands should havethe same radius r , given by the coupling constant m = r − . However, electrons anddown quarks have different rest energies E = ω , and so the relation E = mu = ur implies that either the electron satisfies u (cid:54) = 1, or the down quark satisfies u (cid:54) = 1.That is, either the electron is off-shell or the down quark is off-shell, and so one ofthe two particles must be unstable (Section 3). Our model agrees with experiment,since the down quark is indeed unstable and the electron is stable. However, we cannot explain why electrons and down quarks have different restenergies, or why the only stable quark is the up quark, and so the model requiresfurther development to address stability in the quark sector.10.
Electroweak parity violation
In this section we use the strand model to show that, in the standard model, e R transforms as an SU(2) singlet, and (cid:0) ν e e L (cid:1) transforms as an SU(2) doublet.Consider a diameter, ¯ ψ L ψ R or ¯ ψ R ψ L , of the atom W ± = ( φm ) ¯ ψψ = ( φm )( ¯ ψ L ψ R + ¯ ψ R ψ L ) , in the coupling representation, that is, where the coupling constant φm varies in timeperiodically. As shown in Section 6.1, the two strands on the diameter,¯ α ( t ) = exp β ( t ) ( r cos(¯ ωt ) e + r sin(¯ ωt ) e ) ,α ( t ) = exp β ( t ) ( r cos( ωt ) e + r sin( ωt ) e ) , have opposite angular velocity and equal charge,¯ ω = − ω, q ( ¯ α ) = q ( α ) . We may thus regard the lift of the diameter to the tangent space T β ( t ) ˜ M as a (classical)superposition of ¯ α and α , ( ¯ αα )( t ) := exp β ( t ) (2 r cos( ωt ) e ) , whence as a harmonic oscillator, with charge q ( ¯ αα ) = 2 q ( α ). However, in this formit is clear that the sign of ω is irrelevant to the physical description of the W atom.We want to determine the signs of the angular velocities ω of the strands in thesplit atoms that arise from splitting a W atom. We obtain these signs by the followingthree steps:(i) Since the sign of ω is physically irrelevant in the superposition ¯ αα , we may take ω = | ω | . (This will be the underlying source of the parity violation of neutrinos.)Since α is future-directed, when it splits from the superposition ¯ αα its angular velocityremains | ω | .(ii) We now determine the signs of the individual strands in the two split atoms.Consider an unbarred strand α in one of the split atoms. Recall that the chirality χ ( α ) = sgn( ω ) sgn( α ) of α is determined by which su (2) C subgroup of the complexifiedLorentz algebra so (1 , C its associated element C α = sgn( ω ) J z + i sgn( α ) K z belongs: ψ • ( α ) = (cid:40) ψ L if χ ( α ) = +1 ψ R if χ ( α ) = − α is given by which su (2) C subgroup of so (1 , C the associated element C α = sgn( | ω | ) J z + i sgn( α ) K z SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 29 belongs. Therefore, by the chirality relation (15) we have ψ • ( α ) = ψ L ⇐⇒ sgn( α ) = +1 ,ψ • ( α ) = ψ R ⇐⇒ sgn( α ) = − . (23)Since we are considering the coupling representation, the two strands on a givendiameter of the W atom have the same sign. Thus, by (23), both strands from thediameter ¯ ψ L ψ R have sign −
1, and both strands from the diameter ¯ ψ R ψ L have sign+1. Whence, ψ • ( ¯ α ) = ¯ ψ L ⇐⇒ sgn( ¯ α ) = − ,ψ • ( ¯ α ) = ¯ ψ R ⇐⇒ sgn( ¯ α ) = +1 . (24)(iii) Finally, we use (24) together with the chirality relation (15) to determine thesigns of the angular velocities of the barred strands. We find that neutrinos areonly able to rotate in one direction, and anti-neutrinos are only able to rotate in theopposite direction: W − → ¯ ψ L ψ R ¯ ψ L ⊗ ψ L ψ R ⊗ ¯ ψ R ¯ ψ L ⊗ ¯ ψ R ψ R ⊗ ψ L χ ( B ) = +1 − , +1) ( − , −
1) (+1 , −
1) ( − , +1)sgn( B ) = ( −
1; +1) − − , +1; +1) ( − , +1) ( − , +1; +1) ( − , +1)sgn( ω ( B )) = − − , +1) (+1 , −
1) ( − , −
1) (+1 , +1) B = ¯ ν ↑ e e ↓ µ ↓ ¯ ν ↑ µ τ ↓ ¯ ν ↑ τ W + → ¯ ψ L ψ R ¯ ψ L ⊗ ψ L ψ R ⊗ ¯ ψ R ¯ ψ L ⊗ ¯ ψ R ψ R ⊗ ψ L χ ( B ) = +1 − , +1) ( − , −
1) (+1 , −
1) ( − , +1)sgn( B ) = − −
1; +1) ( − , +1) ( − , +1; −
1) ( − , +1) ( − , +1; − ω ( B )) = − − , +1) (+1 , −
1) ( − , −
1) (+1 , +1) B = ¯ e ↑ ν ↓ e ν ↓ µ ¯ µ ↑ ν ↓ τ ¯ τ ↑ Remark 10.1.
In the field representation of a split atom, say ψ R ⊗ ¯ ψ R or ψ R ⊗ ψ L ,the coupling field φ is bound to the strand α represented by ψ R . In particular, φ has opposite sign to α . However, the signs of the strands in a symmetric atomobtained from the coupling representation are different from the signs obtained inthe field representation (though the total charge of the atom is the same in bothrepresentations). Thus, the sign of the additional charge that arises from the couplingconstant ( φm ) in the coupling representation need not be opposite to that of α . Remark 10.2.
Our model does not admit sterile neutrinos, that is, right-handedneutrinos and left-handed anti-neutrinos which do not interact with W ± atoms. In-deed, each O (3) atom B has • two degrees of freedom from an overall sign of the tuplesgn( ω ( B )) = (sgn( ω α )) α ∈ B of the directions of rotation of its constituent strands α ∈ B ; and • two degrees of freedom from the sign of the total charge q ( B ) = (cid:80) α ∈ B q ( α )of B whenever q ( B ) (cid:54) = 0.However, neutrino atoms have zero total charge, and so a neutrino atom B has onlytwo degrees of freedom, namely the overall sign of sgn( ω ( B )). Furthermore, as wehave just shown, sgn( ω ( B )) alone determines whether B is a neutrino or an anti-neutrino.We thus obtain a derivation of the parity violation of electroweak interactions.Another derivation of parity violation was recently given in [F], and it would inter-esting to understand how the two approaches are related. Our model requires furtherdevelopment to address parity violation in the quark sector.11. A modification to Einsteins equation from the Dirac Lagrangian
Suppose a photon passes through a beam splitter. In the framework of strands,the two path eigenstates of the photon are both physically real or ontic [B5]. Wemay even suppose that the energy of the two eigenstates are equal, and equal tothe initial photon. This does not pose a problem with respect to energy-momentumconservation in the framework of strand spacetime: the two eigenstates are really oneand the same photon sitting at the same point in spacetime M (although at dierentpoints of emergent spacetime ˜ M ), and so the total photon energy is not doubled whenthe photon passes through the beam splitter.However, there is an irreconcilable problem of our two physically real eigenstatesof the photon with Einsteins equation(25) G ab = 8 πT ab given by Bohrs gedankenexperiment: each eigenstate of the photon has energy, andthus produces gravitational radiation as it propagates. But this gravitational ra-diation transmits which-way information, and so the two eigenstates cannot be insuperposition.To remedy this problem, we propose that spacetime curvature is only sourced atapexes, that is, from the mass terms, m ¯ ψ L ψ R and m ¯ ψ R ψ L , of the strand Lagrangian (20). (Recall that in contrast to quantum uctuations, apexesdo not spontaneously occur in a vacuum, but rather only occur to bring o-shell strandparticles on-shell.) The chiral spinors ψ L and ψ R individually have no gravitationalmass since there are no terms of the form(26) m ¯ ψ L ψ L or m ¯ ψ R ψ R SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 31 in the strand Lagrangian.
The absence of the terms (26) implies that the strandparticles represented by ψ L and ψ R do not source gravitation as they propagate. TheEinstein equation on emergent spacetime ˜ M is therefore(27) G ab = 8 π A T ab , where A : ˜ M → { , } is the indicator function defined by A ( x ) := (cid:40) x is an apex0 otherwise . This modification is similar to the semi-classical Einstein equation G ab = 8 π (cid:104) T ab (cid:105) .The problem of Bohrs gedankenexperiment is resolved by the Dirac Lagrangian inthe context of strands. Indeed, the energy density T ab v a v b and momentum densityflow − T ab v a of an observer with four-velocity v a are only defined at points of emergentspacetime ˜ M where an energy-momentum measurement (state reduction) occurs, andthese points precisely form the support of A . The classical Einstein equation is thusrecovered in the classical limit where energy and momentum are defined at each pointof spacetime: A → . A consequence of (27) is the following. Increasing the thermal energy T of anobject, such as a star, results in an increase in the number of interactions (randomcollisions) among its constituent strand particles. These interactions in turn resultin an increase in the frequency of apexes within the object, and so A = A ( T ) isan increasing function of T . Therefore, the hotter an object is, the more it curvesspacetime: as the temperature of the object increases, the indicator function A approaches the identity on the support of the object, A ( T ) T (cid:29) −→ . Conversely, if all the particles making up the object ceased interacting, then theobject would no longer curve spacetime. Consequently, Einsteins equation (25) isrecovered in the high temperature limit, but (27) diers from (25) at low temperature(although the dierence may be extremely small).With the modication (27), the spacetime curvature G ab of ˜ M remains classical. Inparticular, there are no gravitons in this framework. However, there are two possibleconnections between (27) and other theories of gravity: • In our strand model, apexes are responsible both for state reduction of path su-perposition [B5] and gravity. Similar, but dierent, connections between quan-tum state reduction and gravity have been developed by Penrose, Di´osi, andOppenheim [P1, P2, D, O]; in these theories, gravity also remains classical-like. • The source of gravitation in (55) is thermal. It is therefore possible that ourmodel is related to the work of Jacobson [J], Padmanabhan [Pa], or Verlinde[V], among others.
Remark 11.1.
Our model unies electromagnetism with gravity, in the sense that ifwe turn the strength of the electric charge e to zero, so that the positive and negativestrands are no longer attracted to each other, then spacetime curvature, and thusgravity, would disappear.12.
CPT invariance: charge conjugation as a Lorentz transformation
Preliminary: the action of Lorentz transformations on Dirac spinors.
To establish notation, we briefly review the standard derivation of the action of aLorentz transformation Λ ∈ SO (1 ,
3) on a Dirac spinor ψ .To determine the action of Λ on ψ , the Dirac projection ( i /∂ − m ) is required to beinvariant under Λ,(28) Λ( i /∂ − m )Λ − = i /∂ − m. We denote a spinor representation of Λ by S Λ ∈ End( C ), and a spacetime represen-tation by Λ µν ; whence Λ .ψ ( x µ ) = S Λ ψ (Λ µν x ν ) . Thus, (28) implies ( S Λ γ ν )(Λ µν ∂ µ ) = γ µ ∂ µ S Λ = γ µ S Λ ∂ µ . We may therefore obtain S Λ (unique up to a phase) by imposing the constraint(29) S Λ γ ν Λ µν = γ µ S Λ . For example, the parity transformation P = ( P µν ) = diag(1 , − , − , − ψ by(30) P .ψ ( t, x i ) = γ ψ ( t, − x i ) . Charge conjugation as a Lorentz transformation.
Recall the strand La-grangian L = ¯ ψ ( i /∂ − m | ¯ u µ u µ | / ) ψ from Section 6. In Section 7, we found that thephoton atom γ = ¯ ψ L ψ R splits into an electron strand and a positron strand, repre-sented by ψ R and ¯ ψ L respectively. The electron-photon interaction term e ¯ ψ /Aψ inthe standard model Lagrangian is therefore absent from the strand Lagrangian L .In contrast to the term e ¯ ψ /Aψ , the term m ¯ ψ L ψ R in L should not change sign undercharge conjugation, since its coupling constant is mass m , not electric charge e .Geometrically, ˜ M is not fundamentally equipped with a U (1) gauge bundle. In-stead, electric charge arises from the tangent bundle of ˜ M , described in Section 4,and is thus a property of emergent spacetime ˜ M itself. An immediate question, then,is how charge conjugation can be realized in this framework. With only spacetime SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 33 in hand, charge conjugation must somehow be given by a Lorentz transformation.Furthermore, this Lorentz transformation must act on spinors by exchanging particlespinors with anti-particle spinors; a priori, it is not clear whether such a Lorentztransformation exists, or how it could be derived.Consider the four positive energy solutions ( E = + (cid:112) | p | + m ) of the Dirac equa-tion ( i /∂ − m ) ψ = 0 in the Dirac representation: u = N p z E + mp x + ip y E + m , u = N p x − ip y E + m − p z E + m , v = N p x − ip y E + m − p z E + m , v = N p z E + mp x + ip y E + m , where N = √ E + m .In the standard model, charge conjugation C acts by reversing the sign of theelectric charge e ,(31) C ( i /∂ − e /A − m ) C − = i /∂ + e /A − m. Thus C acts on ψ by(32) C .ψ = iγ ψ ∗ . Consequently, C exchanges particle spinors with anti-particle spinors, C .u e ip µ x µ = v e − ip µ x µ , C .u e ip µ x µ = v e − ip µ x µ . (33)Of course, the transformation (32) does not correspond to any Lorentz transforma-tion.In the context of strands, however, the difference between a particle strand and ananti-particle strand lies in the non-uniqueness of tangent vectors (in ˜ M ): a particlestrand α has a future-oriented tangent four-vector τ = ˙ α , and an anti-particle strandhas a past-oriented tangent four-vector τ = − ˙ α , by Definition 4.1. Therefore, since˙ α ( t ) may point in any direction of space, and τ ( t ) = sgn( α ) ˙ α ( t ) , charge conjugation is given by the Lorentz transformation C = ( C µν ) = diag( − , − , − , − . In order for our spinor representation of strands to be consistent, this transformationmust exchange particle spinors with anti-particle spinors.Let us first determine how C acts on a Dirac spinor ψ . Since C is now a Lorentztransformation, we have C ( i /∂ − m ) C − = i /∂ − m. Thus, from (29) we obtain S C γ ν C µν = γ µ S C . Whence S C = γ (times any phase).Therefore(34) C .ψ ( t, x i ) = γ ψ ( − t, − x i ) = (cid:32) II (cid:33) ψ ( − t, − x i ) , where the γ matrix is in the Dirac representation.We find that C does indeed exchange particle spinors with anti-particle spinors,similar to (33): C .u e ip µ x µ = ( γ u ) e i C µν p µ x ν = v e − ip µ x µ , C .u e ip µ x µ = ( γ u ) e i C µν p µ x ν = v e − ip µ x µ . Our model therefore produces a new definition of charge conjugation based solely onthe Lorentz transformation that exchanges a positive strand with a negative strand.
Remark 12.1.
The two spinor transformations S C = iγ and S C = γ both exchangeparticle spinors with anti-particle spinors, although the spinors that are exchanged aredifferent. Specifically, iγ exchanges u and u with v and v respectively, whereas γ exchanges u and u with v and v respectively. Remark 12.2.
The operator γ cannot act as charge conjugation if the term e ¯ ψ /Aψ is included in the Lagrangian, since equation (31) does not hold if S C = γ .12.3. Time reversal without complex conjugation.
Time reversal is the Lorentztransformation T = ( T µν ) = diag( − , , , , which we take to act on the tangent space at a point of emergent spacetime ˜ M . Itis standard to assume that T acts on the spatial velocity vector v of a particle byreversing its sign, since T . v = T . d x dt = d x d ( − t ) = − d x dt = − v . Under this assumption, T acts on the position and momentum operators by(35) T ˆ x T − = ˆ x and T ˆ p T − = − ˆ p. Thus, time reversal acts on complex numbers by complex conjugation:(36) T i T − = T [ˆ x, ˆ p ] T − = (cid:2) T ˆ x T − , T ˆ p T − (cid:3) = [ˆ x, − ˆ p ] = − i. Consequently, T acts on the Dirac projection by complex conjugation,(37) T ( i /∂ − m ) T − = ( i /∂ − m ) ∗ . From (37), and the fact that γ is pure imaginary, it follows that T acts on ψ by T .ψ = γ γ ψ ∗ . SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 35
However, we claim that time reversal does not actually reverse the sign of thevelocity vector v , and so conjugation of the momentum operator ˆ p by T does notreverse the sign of ˆ p .Indeed, consider the worldline x µ = x µ ( t ) of a particle. The particle has four-velocity ( ˙ x µ ) = γ (1 , v ) , where γ = (1 − | v | ) − / . Observe that T leaves v unchanged:( T µν ˙ x ν ) = γ ( − , v ) . In other words, the output of the spatial component of the time derivative ˙ x µ is thespatial vector v , and so the spatial component of the Lorentz transformation T actson v , not the time component. Therefore the particle’s spatial momentum, p = γm v ,is also left unchanged by T , ( T µν p ν ) = γm ( − , v ) . Consequently, we have T ˆ p T − = ˆ p, in contrast to (35). Whence (37) does not hold; instead, the Dirac projection remainsinvariant under time reversal, as it is a Lorentz transformation: T ( i /∂ − m ) T − = i /∂ − m. Therefore, applying (29) we find(38) T .ψ ( t, x i ) = γ γ γ ψ ( − t, x i ) . Remark 12.3.
The Lorentz transformation that does reverse the direction of p ischarge conjugation C , as we found in Section 12.2. Whence C ˆ p C − = − ˆ p . But C alsoreverses the sign of x , and so C ˆ x C − = − ˆ x . Thus C i C − = i by (36), and therefore C does not induce complex conjugation either.12.4. CPT invariance.
In Section 12.2, we showed that charge conjugation C of astrand is given by the Lorentz transformation(39) C = diag( − , − , − , − . Furthermore, in Section 12.3, we showed that time reversal T does not induce complexconjugation.To summarize, charge conjugation (34), parity (30), and time reversal (38) act onDirac spinors in the strand preon model by C .ψ ( t, x i ) = γ ψ ( − t, − x i ) , P .ψ ( t, x i ) = γ ψ ( t, − x i ) , T .ψ ( t, x i ) = γ γ γ ψ ( − t, x i ) . (40) Note that if we used the signature ( − , , ,
1) instead of (1 , − , − , − S P and S T would beswapped. Each of these transformations act trivially on the Dirac projection ( i /∂ − m ).There are two important consequences of (39) and (40): • The Lorentz group O (1 , O (1 , /SO + (1 ,
3) has four elements. Using (39), this quotientis generated by C , P , and T : O (1 , /SO + (1 ,
3) = (cid:104) C , P , T (cid:105) ∼ = Z × Z . Each of 1, C , P , and T belong to a different one of the four connected components of O (1 , • The product
CPT , in the spacetime representation, is simply the identity:(41) C µρ P ρλ T λν = g µν = δ µν ∈ SO + (1 , . Furthermore, the product
CPT , in the spinor representation, is proportional to theidentity:(42) S C S P S T = γ γ ( γ γ γ ) = − i ( γ ) = − i. Note that (42) is consistent with (41) since g µν may be taken to act on a Dirac spinorby multiplication by an arbitrary phase e iθ , by (29).By (41) and (42), CPT invariance of the strand Lagrangian L trivially holds. Remark 12.4.
The relationships between C , P , and T that we have obtained cannotoccur in the standard model, because in the standard model charge conjugation doesnot correspond to a Lorentz transformation. Remark 12.5.
The relation (
CPT ) µν = δ µν in (41) is really just a different incarnationof the chirality relation sgn( α ) χ ( α ) sgn( ω ) = 1from (15), resulting from the correspondences C ∼ sgn( α ) , P ∼ χ ( α ) , T ∼ sgn( ω ) . The correspondence P ∼ χ ( α ) holds because χ ( α ) is the handedness, left or right, ofthe ordered basis { t , n , b } (shown in Section 5), and parity P flips the handedness ofthis basis: P . { t , n , b } = {− t , − n , − b } . Furthermore, the correspondence T ∼ sgn( ω ) holds because t and ω only appear as theproduct ( ωt ) in the parameterization of circular strands in (3), and ( − ω ) t = ω ( − t ). Remark 12.6.
A rotation by 2 π in spacetime corresponds to the Lorentz transfor-mation ( PT ) . Furthermore, by (40),( S P S T ) = ( γ γ γ γ ) = − . We therefore obtain the elementary fact that a spacetime rotation of a spinor ψ in C is trivial if and only if the rotation is a multiple of 4 π . SPECTS OF THE STAND. MODEL & QUANT. GRAVITY FROM STRAND SPACETIME 37
The spin-statistics connection for strands
The following derivation is similar to Schwinger’s heuristic argument for the spin-statistics connection [Sch], with CP in place of T and CPT . However, in the frame-work of strands, the complications due to time (e.g., spacelike separation of the twoparticles; rotation to Euclidean spacetime) do not arise.Consider two strands α , ˜ α of equal radius r . Denote by β , ˜ β their respective centralworldlines in ˜ M , and suppose that there are points along β and ˜ β , say β ( s ) and ˜ β (˜ s ) , that are causally connected. Further suppose that the strands have equal chirality,say left-handed, with spinor representations ψ L ( β ( s )) and ψ L ( ˜ β ( s )) . By a possible local change of coordinates, we may suppose that x µ := ˜ β µ (˜ s ) = − β µ ( s ) . Furthermore, by the indistinguishability of swapping two identical particles, we have(43) PT . ( ψ L ( − x µ ) ⊗ ψ L ( x µ )) = ψ L ( − x µ ) ⊗ ψ L ( x µ ) . Thus, using parity and time reversal of strands, we find (suppressing µ ) ψ L ( − x ) ⊗ ψ L ( x ) (i) = PT . ( ψ L ( − x ) ⊗ ψ L ( x ))= ( PT .ψ L ( − x )) ⊗ ( PT .ψ L ( x )) (ii) = ( − iγ ) ψ L ( x ) ⊗ ( − iγ ) ψ L ( − x )= iψ L ( x ) ⊗ iψ L ( − x )= − ψ L ( x ) ⊗ ψ L ( − x ) . (44)where ( i ) holds by (43), and ( ii ) holds by (40). Similarly, for a diameter ¯ ψ L ψ R wefind ( ¯ ψ L ψ R )( − x ) ⊗ ( ¯ ψ L ψ R )( x )= PT . (( ¯ ψ L ψ R )( − x ) ⊗ ( ¯ ψ L ψ R )( x ))= ( ¯ ψ L ( − iγ )( − iγ ) ψ R )( x ) ⊗ ( ¯ ψ L ( − iγ )( − iγ ) ψ R )( − x )= ( ¯ ψ L ψ R )( x ) ⊗ ( ¯ ψ L ψ R )( − x ) . (45)In field-theoretic terms, an excitation of the field ψ L at β µ ( s ) is the same fieldexcitation at β µ ( s ) since the two points α µ ( s ) and α µ ( s ) , by virtue of being joined by the strand α , are the same point in spacetime M . There-fore (44) and (45) hold along the entire two central worldlines β and ˜ β , ψ L ( β ) ⊗ ψ L ( ˜ β ) = − ψ L ( ˜ β ) ⊗ ψ L ( β ) , ( ¯ ψ L ψ R )( β ) ⊗ ( ¯ ψ L ψ R )( ˜ β ) = ( ¯ ψ L ψ R )( ˜ β ) ⊗ ( ¯ ψ L ψ R )( β ) , (46)and similarly for L ↔ R . In particular, the central worldlines of two strands of equalchirality and equal radius cannot intersect: for x ∈ ˜ M we have ψ L ( x ) ⊗ ψ L ( x ) = 0 and ψ R ( x ) ⊗ ψ R ( x ) = 0 . This is the Pauli-exclusion principle for strands. Furthermore, by (46), the centralworldlines of diameters are allowed to intersect. In particular, (46) implies that(circular) strands are fermionic, whereas diameters are bosonic. We therefore obtaina spin-statistics connection for strands.14.
Future directions: scattering with strands
We assume that the standard Feynman rules hold for propagators and vertices, asan effective field theory, for computing cross sections and decay rates. However, (atleast) two new constraints on the path integral arise in our strand preon model.To describe these constraints, consider a scattering event with fixed incoming andoutgoing particles. In this event, all the different configurations of internal lines(strands) that exist in superposition, exist together in spacetime M . By the spin-statistics connection for strands (Section 13), the worldlines of strands with the samechirality cannot intersect unless at least one of the strands is paired with anotherstrand in a diameter of an atom, making the pair bosonic. Furthermore, a strandparticle cannot simultaneously both terminate at an apex and continue to propagate.We thus expect severe restrictions on both(i) the possible Feynman diagrams that may exist in superposition; and(ii) the limits of integration (cid:82) (cid:82) d x d p of the Lagrangian density.Scattering amplitudes obtained from strands should therefore differ from thoseobtained from the full path integral Z = (cid:82) D ψe i (cid:126) S [ ψ ] . These constraints could poten-tially eliminate certain (ultraviolet) divergences in quantum field theory. We leave aformulation of these constraints for future work. Acknowledgments.
The author would like to thank Gary Beil, Neil Gillespie, andVinesh Solanki for valuable discussions. The author was supported by the AustrianScience Fund (FWF) grant P 30549-N26.
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