Foundations of Physics manuscript No. (will be inserted by the editor)
Lorentz Symmetry and the Curved Spacetime Dirac Equations G. G. Nyambuya
Received: date / Accepted: date
Abstract In a part of our previous works, we have argued (suggested) that it should – in-principle – be possible for a massive particle to travel at the vacuum speed of light c if it rest-mass is a function of momentum. In-principle, this allows one to explain why – say – neutrinos have a non-zero rest-mass and travel at the vacuum speed of light. We explore this idea in the context of the curved space time Dirac equation which we proposed in some of our earlier works. Additionally, using a hypothetical example, we demonstrate why this idea of a momentum dependent rest-mass must be taken seriously as one may be able to justify (explain/ account for) quantization of momentum and as-well the nonobservability of negative energy particles.
1 Introduction About eight years ago (Nyambuya 2008), we published in the present journal what we believe are legitimate Curved Spacetime (CST) versions of the Dirac (1928a,b) equation. In the present reading, we explore the resulting energy equations of these CST-Dirac equations, that is, whether or not they result in the violation of Lorentz Symmetry. We argue that these CST-Dirac equations do obey Lorentz symmetry provided that the rest-mass of the particle depends on its momentum. As is well known, Lorentz symmetry is a sacrosanct cornerstone of Einstein (1905)’s Special Theory of Relativity (STR). Lorentz symmetry is the postulate (hypothesis) that G. G. Nyambuya National University of Science & Technology, Faculty of Applied Sciences – Department of Applied Physics, Fundamental Theoretical Astrophysics Group, P. O. Box AC 939, Ascot, Bulawayo, Republic of Zimbabwe. Tel.: +263-77-7960-452 E-mail:
[email protected]
all observers in the Universe – at anytime and anywhere – are going measure exactly the same numerical value for the speed of light in vacuum (c = 2.99792458 × 108 ms−1 ), independent of the photon’s energy (and or momentum; or any other physical attribute of the photon). Somewhat in violation of this symmetry, recent astronomical observations of Gamma-Ray Burst (GRB) events have shown that γ -rays of different energies all emanating from the same GRB event, these γ -rays have been observed to arrive at the telescope at different times depending on the γ -ray’s energy (see e.g. Amelino-Camelia et al. 1998; Amelino-Camelia 2001, 2002; Boggs et al. 2004; Cardone et al. 2006; Jacobson et al. 2006; Mirabal et al. 2006; Aharonian et al. 2008; Amelino-Camelia 2013; Vasileiou et al. 2013). This suggests (amongst other possibilities) a violation of Lorentz symmetry. These GRBs have become a major focal point for experimental searches for signs of QG-effects. Appart from GRBs, effort has (see e.g. Williams et al. 1971) and is also under-way to investigate possibilities of Lorentz violations on the quantum (see e.g. Mattingly 2005; Cardone et al. 2006; Jenkins 2004). Additionally, some effort is also being made in the Solar system, on the galactic (see e.g. Jacob and Piran 2008; Stecker and Glashow 2001) and cosmological scales. At present, one can safely that, more than in any field of scientific endeavour, Lorentz violations are a major derive for those seeking to tie together Einstein (1916)’s General Theory of Relativity (GTR) and Quantum Theory (QT) into a Quantum Gravity (QG) theory. The majority of these quantum gravity efforts seek to find deviations in Einstein (1905)’s momentum-energy dispersion relation, namely: E 2 − p2c2 = m20 c4 ,
(1)
where E is the energy of the particle, p and m0 are this same particle’s momentum and rest-mass respectively. Any devi-
2
G. G. Nyambuya
ations, will [even in the case (m0 = 0)], lead to an energy dependant speed of the photon. In the present reading – as already said in the opening paragraph, we shall argue that the proposed CST-Dirac equations (Nyambuya 2008) do obey Lorentz symmetry. This is important because two of these CST-Dirac equations lead to a deformed momentum-energy dispersion relation. From the prevailing QG-approach, this would mean that these equations – must too – lead to Lorentz symmetry violation.
2 Quantum Gravity: Deformed Dispersion Relation Present attempts at merging QM and GTR involve the modification of Einstein (1905)’s momentum-energy dispersion relation (1). These attempts propose a modification (deformation) of this equation (1), so that, it is now given by: E 2 − p2 c2 = m20 c4 + f (p, E : EQG ),
(2)
where [ f = f (p, E : EQG )] is a function that depends on the E and p and the crucial aspect in this modifications is the introduction of an observer independent invariant energy-scale EQG known as the quantum gravity energy scale and this scale is related to an invariant scale-length (ℓ p = h¯ c/EQG ); where (¯h = 1.06 × 10−35 Js), is Planck’s normalised constant. This invariant scale-length is usually assumed to co1 incide with the Planck scale-length [ℓ p = (G¯h/c3 ) 2 ], where (G = 6.667 × 10−11 kg−1 m3 s−2 ) is Newton’s universal constant of gravitation: in this case (ℓ p = 5.11 × 10−36 m). The introduction of an independent invariant scale-length was first introduced by Pavlopoulos (1967, 1969) where it was assumed to coincide with the atomic scale 10−15m. From this deformed momentum-energy equation, we can calculate the group velocity vg of the associated wave-packet. We know that:
vg =
∂E . ∂p
(3)
Thus, differentiating the deformed momentum-energy (2) with respect to p, we obtain after some basic algebraic operations: β
z "
}| #{ 2 pc/E + (1/2cE)∂ f /∂ p + m0 c /E ∂ m0 c/∂ p vg = c. 1 − (1/2E)∂ f /∂ E
are being sought e.g. in the gamma-rays bursts (Pavlopoulos 2005; Xiao and Ma 2009; Kislat and Krawczynski 2015). If no signal can exceed c, then, the factor β is such that (vg ≤ c), this implies: 1 + (1/2pc2)∂ f /∂ p + (m0 c/pc) ∂ m0 c/∂ p pc. (5) E≥ 1 − (1/2E)∂ f /∂ E
In the usual Einstein momentum-energy equation, for f , we have ( f = 0). As is well known, for this equation i.e. equation (1), if (m0 = 0), we will have (vg = c), that is, such particles [with (m0 = 0)] will propagate at the speed of light c. This has raised the age-old question of whether or not massive photons that travel at the usual speed of light c can exist (see e.g. Robles and Claro 2012, for a pedagogical exposition). As argued in the reading Nyambuya (2014), in the case of massive photons satisfying Einstein (1905)’s momentumenergy dispersion relation (1), if the non-zero rest mass of such photons is not a function of the momentum, then, they will certainly travel at speeds less than the speed of light c because: # ∂ m20 c4 + 1 pc = pc, E> ∂ (p2 ) "
(6)
since [∂ m20 c4 /∂ p2 = 0] and (E 2 − p2 c2 = m20 c4 > 0). According to the foregoing, for massive photons to travel at the speed of light, two things must be satisfied, that is, the momentum of their rest mass must be such that derivative [∂ m20 c4 /∂ p2 6= 0], and the second is that – the energy E of such photons must be such that: "
# ∂ m20 c4 E= + 1 pc > 0. ∂ (p2 )
(7)
In a nutshell, what the above thesis implies (or is pointing to) is that the value of the rest mass of a particle, that is, whether it is zero or non-zero, this is not enough to decide whether or not a particle will travel at the speed of light or less than this speed. What is important is the variation of the particle’s rest mass with respect to its momentum – this is the final arbiter in the matter.
(4) 3 Hypothetical Momentum Dependent Rest-mass
The above implies that vg = vg (E, p : EQG ). In the case of light, what (4) implies is that the speed of light may dependent on its energy and momentum and such deviations
As a way of demonstrating the power of having m0 depend on the particle’s momentum, we here consider a very in-
3
Lorentz Symmetry and the Curved Spacetime Dirac Equations
teresting – albeit; hypothetical momentum dependent restmass. Suppose:
π , m0 = m exp − q 2 1 − p2c2 /EQG
(8)
where m is a fundamental constant with the units of mass and π has been inserted for convenience. According to (4), if the rest-mass of a particle is given as in (8), then, in-order for such a particle to travel at the speed of light c, its energy must be related to its momentum as follows:
E = 1 +
c2
m0 EQG
2
1−
p 2 c2 2 EQG
!−3/2
pc.
(9)
If [(pc, m0 c2 ) ≪ EQG ], then, to first order approximation, we will have (E ≃ pc) for these massive particles travelling at the speed of light. Further, suppose we have two energy scales EQG1 and EQG2 such that the energy, E, of particle can only lay in the region: (EQG1 ≤ E ≤ EQG2 ), then, we can introduce the function: v u 2 2 2 u p c /EQG1 − 1 . m0 = m exp −iπ t 2 1 − p2c2 /EQG2
(10)
This is a very interesting function because – If: v u 2 2 2 u p c /EQG1 − 1 πt = π n, where (n = ±1, ±2, ±3, . . . ) 2 1 − p2c2 /EQG2
(11)
then, m0 will be real. This must result in particles that we are used too, that is, particles that travel at the speed of light of less than that. For these particles, given the condition (pc ≪ EQG2 ), it follows that to first order approximation, we will have: p EQG1 2 p = ± 1+n ≃ np. c
(12)
On the other hand – If: v u 2 2 2 u p c /EQG1 − 1 π πt = + π n, where (n = ±1, ±2, ±3, . . . ) 2 2 1 − p2c2 /EQG2 (13)
then, m0 will be an imaginary quantity. This must result in particles that we are not used too, that is, particles that travel at speeds greater than the speed of light. For these particles, given the condition (pc ≪ EQG2 ), it follows that to first order approximation, we will have: q 1 EQG1 2 p = ± 1 + (1 + 2n) /4 ≃ + n p. c 2
(14)
If a particle with imaginary mass is a tachyon (vg > c), and that with a real mass is a bradyon (vg < c), then, in the hypothetical case that we have just considered, it is seen that the quantization of momentum (into particles with integral and half-integral momentum) is a necessary condition to separate tachyons from bradyons (or vice-versa). At this point, one may therefore wonder; is this the way Nature has chosen so as to put a dividing veil between the worlds? 4 Curved Spacetime Dirac Equations Here at the outset, it is perhaps important that we mention that, there are several curved spacetime versions of the Dirac equation (cf. Alhaidari and Jellal 2014; Arminjon and Reifler 2013, 2010; Pollock 2010; Arminjon 2008; Weyl 1927a,b; Fock 1929), each with its unique taste and flavour in how it is arrived at. In our humble and modest view; save for the introduction of a seemingly mysterious four vector potential Aµ , what makes the curved spacetime version of the Dirac equations presented in Nyambuya (2008) stand-out over other attempts is that the method used in arriving at these curved spacetime Dirac equations (Nyambuya 2008) is exactly the same as that used by Dirac (1928a,b). As will be demonstrated shortly, this method used in Nyambuya (2008) appears to us as the most straight forward and logical manner in which to arrive a curved spacetime version of the Dirac equation. All that has been done in Nyambuya (2008) is to decompose the general Riemann metric g µν in a manner that allows us to apply Dirac (1928a,b)’s prescription at arriving at the Dirac equation. Dirac (1928a,b)’s original equation is arrived at from the Einstein momentum-energy dispersion equation (1, which we can write in tensor notation as: (ηµν pµ pν = m20 c4 ), where ηµν is the usual Minkowski metric, (pµ , m0 ) are the four momentum and rest mass of the particle in question respectively and c is the usual speed of light in a vacuum. In curved spacetime, we know that this equation (ηµν pµ pν = m20 c4 ) is given by the equation (gµν pµ pν = m20 c4 ) where g µν is the general Riemann metric of a curved spacetime. If a curved spacetime version of the Dirac equation is to be derived, it must be derived from this fundamental energy equation (gµν pµ pν = m20 c4 ), in much the same way the flat spacetime Dirac equation is derived from the fundamental equation: (ηµν pµ pν = m20 c4 ). Dirac derived his equation by
4
G. G. Nyambuya
taking the ‘square-root’ of the equation (ηµν pµ pν = m20 c4 ). It is a fundamental mathematical fact that a two rank tensor (such as the Riemann metric tensor g µν ) can be written as a sum of the product of a vector Aµ , i.e. (gµν = σµν Aµ Aν ) where σµν is set of co-efficient taking the values (0, ±1). For this metric tensor (gµν = σµν Aµ Aν ), we are going to have three cases.
4.4 New Compact Form of Energy-Momentum Equation These three equations (15, 16 and 17) can compactly be written as:
(a)
gµν
z
}| { (a) (a) 2 4 m0a c I4 = Aµ γµ Aν γν pµ pν .
4.1 Case (a = 1)
(18)
For the avoidance of confusion with the usual Einstein sum(a) mation convention, the object Aµ γµ is to be treated as sinThe first is the case where the off diagonals are zero i.e. gle object, that is to say, the reader can say write this as (σµν = 0) for (µ 6= ν ) and the diagonals are such that (σ00 = 1) (a) (a) (a) (a) (a) hµ = Aµ γµ so that (gµν = hµ hν ). and (σ = −1), in which event we will have: ii
m201 c4 = A0 A0 p0 p0 − A1 A1 p1 p1 − A2 A2 p2 p2 − A3 A3 p3 p3 .
(15)
The coefficients of σµν are in this case set such that the values of σµν coincide with the Minkowski metric so that whenever (Aµ = 1) for all µ , gµν is the Minkowski metric.
4.2 Case (a = 2) In the second case, we are going to have the off-diagonals being no-zero i.e. (σµν = 1) for (µ 6= ν ) and the diagonals are such that (σ00 = 1) and (σii = −1), in which event, we are going to have:
4.5 New Dirac Equations As one can verify for themselves that, the ‘Dirac-squareroot’ of (18) is given by: i h (a) i¯hAµ γµ ∂ µ − m0ac ψ = 0.
Actually, it is not a difficult exercise to show that multiplication of (19) from the left hand-side by the conjugate operator: h i† µ† i¯hAµ γ(a) ∂µ − m0ac , leads us to the Klein-Gordon equation: gµν ∂ µ ∂ ν ψ =
m202 c4 = A0 A0 p0 p0 − A1 A1 p1 p1 − A2 A2 p2 p2 − A3 A3 p3 p3 + 2A0 A1 p0 p1 + 2A0 A2 p0 p2 + 2A0 A3 p0 p2 +
.
(16)
2A1 A2 p1 p2 + 2A1 A3 p1 p3 + 2A2 A3 p2 p3
4.3 Case (a = 3)
m0a c2 h¯
2
ψ,
provided (∂µ Aµ = ∂ µ Aµ = 0); this afore-stated condition (∂µ Aµ = ∂ µ Aµ = 0) should be taken as a gauge condition restricting this four vector.
(a)
Written as a stand-alone entity, the metric gµν is such that:
(a)
m203 c4 = A0 A0 p0 p0 − A1 A1 p1 p1 − A2 A2 p2 p2 − A3 A3 p3 p3 +
2A1 A2 p1 p2 + 2A1 A3 p1 p3 + 2A2 A3 p2 p3
4.6 New Gamma-matrices
In the third case, we are again going to have the off-diagonals being no-zero such that (Aµ 7→ −Aµ ), in which event, we are going to have:
−2A0 A1 p0 p1 − 2A0 A2 p0 p2 − 2A0 − A3 p0 p2 +
(19)
.
(17)
gµν =
1 2
=
1 2
o n (a) (a) Aµ γµ , Aν γν
n o (a) (a) γµ , γν Aµ Aν , (a)
= σµν Aµ Aν
(20)
5
Lorentz Symmetry and the Curved Spacetime Dirac Equations (a)
(a)
where σµν are 4 × 4 matrices such that σµν =
1 2
and γ (a) -matrices1 are defined such that:
n o (a) (a) γµ , γν
respectively. From the n foregoing, itofollows that the writing (a) (a) (a) (a) 1 of g µν as g µν = 2 Aµ γµ , Aν γν is to be accepted as a (a)
(a)
γ0
(a) γk
=
=
1 2
I2 0 0 −I2
and, (21)
√ iλ 1 + λ 2σ k 2λ I2 √ , −2λ I2 −iλ 1 + λ 2σ k
where I2 is the 2 × 2 identity matrix, σ k is the usual 2 × 2 Pauli matrices and the 0’s are 2 × 2 null matrices and a = (1, 2, 3) such that for: 1, then (λ = 0) : Flat Spacetime. a = 2, then (λ = +1) : Positively Curved Spacetime. 3, then (λ = −1) : Negatively Curved Spacetime. (22) The index “a” is not an active index as are the Greek indices. This index labels a particular curvature of spacetime i.e. whether spacetime is flat2 , positive or negatively curved.
(1)
(2)
Written in full, the three Riemann metric tensors gµν , gµν and
are given by:
A0 A0 0 0 0 i 0 −A A h 0 0 (1) 1 1 , gµν = 0 0 −A2 A2 0 0 0 0 −A3 A3
(23)
A0 A0 A0 A1 A0 A2 A0 A3 i A A −A A A A A A h (2) 1 0 1 1 1 2 1 3 g µν = A2 A0 A2 A1 −A2 A2 A2 A3 , A3 A0 A3 A1 A3 A2 −A3 A3
(24)
A0 A0 −A0 A1 −A0 A2 −A0 A3 h i −A A −A A A A A1 A3 (3) 1 0 1 1 1 2 g µν = −A2 A0 A2 A1 −A2 A2 A2 A3 , −A3 A0 A3 A1 A3 A2 −A3 A3 ,
(25)
1
Just to emphasis once more: in equation (19) above, the term µ Aµ γ(a) must be treated as a single object with one index µ . This is µ
4.8 General Spin Dirac Equations As it stands, equation (19) would be a horribly complicated equation insofar as its solutions are concerned because the vector Aµ is expected to be a function of space and time i.e. Aµ = Aµ (r,t). Other than a numerical solution, there is no foreseeable way to obtain an exact solution is if that is the case. However, in the readings Nyambuya (2009, 2013), we found a way round the problem; we realised that transformations: Aµ (r,t) 7→ sµ φ(r,t) , m0a 7→ m0a φ(r,t)
4.7 Configurations of the Metric Tensor
(3) gµν
legitimate mathematical fact for as long as gµν is a tensor. Since Aµ is a vector and the γ (a) -matrices are all constant (a) matrices, gµν is a tensor. Alternatively, it follows that the energy-momentum equation, g µν pµ pν = m20 c4 , can now be written as: o 1n (a) (a) Aµ γµ , Aν γν pµ pν = m20a c4 . 2
(26)
where φ(r,t) is scaler and sµ is vector that does not dependent on the space and time coordinates. These transformations (26) lead to equation (19) to now be given by: i h (a) i¯hsµ γµ ∂ µ − m0a c ψ = 0,
(27)
As shown in the readings Nyambuya (2009, 2013), the quantity sµ is defined such (s0 = 1), (sk = 0, ±1, ±2, ±3, . . . etc). In equation (27), the vector Aµ has completely disappeared from our midst thus drastically simplifying the resultant equation in the process. The quantity sk has been shown to be the representative of the spin of the particle, that is to say, the spin s i such that s = sk h¯ σ k /2.
5 Energy Solutions As first shown in the reading Nyambuya (2008), the resulting energy equation for the CST-Dirac equations, is:
µ
what this object is. One can set Γ(a) = Aµ γ(a) . The problem with this µ
setting is that we need to have the objects, Aµ and γ(a) , clearly visible in the equation. 2 By flat, it here is not meant that the spacetime is Minkowski flat, but that the Riemann metric has no off diagonal terms. On the same footing, by positively curved spacetime, it meant that Riemann metric has positive off diagonal terms and likewise, a negatively curved spacetime, it meant that Riemann metric has negative off diagonal terms.
Ea2 − p2 c2 − λa c
"
3
∑ pi
i=1
#
Ea + λa2 c2
"
3
3
∑ ∑ pi p j
i=1 j=1
#
= m20a c4 . i6= j
(28) This energy-equation represents a particle of spin h¯ /2 i.e. (sk = 1). This is the equation that we are going to analysis.
6
G. G. Nyambuya
We wish to find out if this equation does lead to a violation of Lorentz symmetry. The result obtained hold for all other case for which (|sk | > 1). Following the scheme used in QG, we can identity the function f as:
f a = λa c
"
3
∑ pi
i=1
#
Ea − λa2 c2
"
3
3
∑ ∑ pi p j
i=1 j=1
#
.
(29)
i6= j
The subscript-a in f has been inserted so as to identity the different cases for the curvature states [a = (1, 2, 3)]. Certainly, this equation (28), is a deformed momentum-energy equation with fa as given (29) being the deformation function (term). Within the realms, understanding, context and present approach in QG, this equation (28) must lead to Lorentz symmetry violation because of fa .
Differentiating fa with respect to p and making use of the relations (30), and thereafter doing some rearrangement, we will have:
1 ∂ f a λa p = 2cEa ∂ p 2
"
3
∑
i=1
# " # 1 λa2 pc 3 3 p j pi − ∑ ∑ pi + p j pi 2Ea i=1 j=1
.
i6= j
(31) and: 1 ∂ fa λa c = 2Ea ∂ Ea 2Ea
"
3
∑ pi
i=1
#
.
(32)
Given that the group velocity is given by vg|a = ∂ Ea /∂ p , it follows that the group velocity will be given by:
6 Resultant Group Velocity In the present section, we shall calculate the resulting group velocity from the momentum-energy equation (28). We know that (p2 = p21 + p22 + p23); from this it follows that:
∂p ∂ pi pi p = ⇒ = . ∂ pi p ∂p pi
(30)
i h (pc/Ea ) + (λa p/2) ∑3i=1 1/pi − λa2 pc/2Ea ∑3i=1 ∑3j=1 (p j /pi + pi /p j )i6= j + m0a c2 /Ea (∂ m0a c/∂ p) vg|a . = c 1 − (λa/2Ea ) ∑3i=1 pi c
In-order that the denominator in (33) is greater than zero so as to avoid the clear singularity, for the cases [a = (2, 3)], we must have: 2 Ea > λa
3
∑ pi c
i=1
!
.
(34)
Further, in-order that (vg|a > 0), we must have: If we assumed that the rest-mass is not a constant but is dependent on the particle’s momentum, and that (vg|a ≤ c) then, for the cases [a = (2, 3)]: υa (p)
z
i { h }| 1 + λa p ∑3i=1 1/pi − 21 ∑3i=1 ∑3j=1 (p j /pi + pi /p j )i6= j pc, Ea ≥ 1 − (m0a c/p) (∂ m0a c/∂ p)
(35)
so that (Ea ≥ υa pc): υa is such that (υa2 = 1 + fa /pc).
(33)
7 General Discussion While the CST-Dirac equations lead to deformed momentumenergy dispersion relations, we have herein demonstrated that these equations indeed uphold Lorentz symmetry. From current efforts at a QG-theory, deformed momentum-energy dispersion relations are taken as indispensable tools to probe possible quantum gravity effects. The ‘curious’ hypothetical momentum dependent restmass analysed in §(3) hints at something subtle, namely that, while this momentum dependence of the rest-mass can be used to attain massive particles that travel at the speed of light, this same idea may be useful in explain the quantization of energy and momentum. We have not derived this momentum dependence from an fundamental principle, it is only hypothetical and can not be taken very seriously as a credible theory. Be that is may be, hypothetical it may be, this suggestions hints at something really subtle and interesting – which Nature may or may not have employed to
Lorentz Symmetry and the Curved Spacetime Dirac Equations
attain quantization. It would be worthy while if a fundamental basis for such a relation is found.
8 Conclusion We hereby make the following conclusion: that, the proposed curved spacetime Dirac equations (Nyambuya 2008) do obey Lorentz symmetry.
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