Dequantization of Dirac equation: emergence of spin and antimatter in special relativity Alberto C. de la Torre∗ Universidad Nacional de Mar del Plata Argentina October 3, 2018 It is shown that the existence of spin and antimatter are a prediction of special relativity together with a postulate of a linear relation between energy and momentum. No reference is made to any quantization postulate.
Keywords: special relativity, Dirac equation, spin, antimatter
I.
INTRODUCTION
Following the success of Schr¨odinger’s equation, the relativistic extension was made leading to Klein-Gordon equation. This equation presented several difficulties, like negative probabilities, whose solution implied some drastic moves. So, Dirac proposed a relativistic quantum mechanic equation[1] that made mainly two spectacular predictions: the existence of spin and antimatter. Since then, these two features of reality were considered to have been discovered by relativistic quantum mechanics. However, the analysis presented here shows that spin and antimatter are a consequence of relativity and the postulation of a linear energy-momentum relation, without any reference to a quantization postulate. We will develop here a classical (non quantum) relativistic dynamic containing these two important features. In a sense, we dequantize Dirac’s equation and we see that the remaining geometrical and algebraic structure contains spin and antimatter. This theoretic ∗
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2 result, that spin and antimatter have not a quantum origin, is relevant because quantum mechanics has several interpretation difficulties that may result in drastic modifications of the theory that will therefore leave the concepts of spin and antimatter untouched.
II.
ENERGY MOMENTUM RELATIONS
Energy E = p0 and momentum p = (p1 , p2 , p3 ) are the fundamental dynamic variables, sufficient conditions for the existence of a particle. This scalar and vector quantity are not independent and must satisfy some energy-momentum relation F (E, p) = 0. Since energy and momentum are the generators of time and space translations, this energy-momentum relation fixes a space-time equation of motion for the particle. For example, the non relativistic energy-momentum relation for a particle of mass m moving under the effect of a force F derived from a potential V (x), E=
p2 , 2m
(1)
with p2 = p · p = pk pk , leads to Newton’s second law and to the momentum dependence on velocity. To prove this, consider the Hamiltonian H(x, p) = E +V = p2 2m
+ V (x) and with Hamilton’s equations ∂H ∂V dpk =− k =− k dt ∂x ∂x k k dx ∂H p = k = dt ∂p m
(2) (3)
we get dp =F, dt
(4)
p=mv ,
(5)
1 E = mv 2 . 2
(6)
In a similar way, the relativistic energy-momentum relation m2 = pµ pµ = E 2 − p2 ,
(7)
3 (where we use four-vector notation µ = 0, 1, 2, 3 , k = 1, 2, 3, c = 1 and the metric gµν = Diag(1, −1, −1, −1)) generates the relativistic equations of motion dp =F, dt
(8)
p=γmv ,
(9)
E=γm,
(10)
√ where γ = ( 1 − v 2 )−1 is the Lorentz factor. In relativistic quantum mechanics, the equation of motion arising from the quadratic energy momentum relation (7), implied some difficulties that lead P. M. Dirac to postulate a linear equation of motion that could be thought to come from the linear energy momentum relation m = γ µ pµ = γ 0 E − γp ,
(11)
with appropriate coefficients γ µ (not to be confused with the Lorentz factor). Both equations, the quadratic (7) and the linear (11) must be satisfied. Clearly this is not possible with numbers for γ µ . Squaring equation (11) and comparing with (7) we find that the coefficients must satisfy an algebra defined by the anti-commutator {γ µ , γ ν } = γ µ γ ν + γ ν γ µ = 2g µν .
(12)
Both energy momentum relations, the quadratic and the linear, lead to the same equations of motion but the linear equation may have additional consequences not following from the quadratic one (notice that (11) implies (7) but not the opposite). The linear energy-momentum relation in Eq.(11) can be called a dequantized Dirac equation because it concerns dynamic variables and not Hilbert space operators. Its interest arises not only as solution to some difficulties of the quadratic relation in quantum mechanics. A constant velocity
dx , dt
the essence of the iner-
tia principle, implies a linear relation between time and position coordinates of a particle, components of the four-vector xµ . Another important four-vector for a particle is pµ and we may apply a being-becoming symmetry principle[2] that relates position and momentum, and wonder if a constant
dp , dE
that would imply a linear
(E, p) relation, is a physically relevant concept. A linear relation among coordinates
4 (t, x) is characteristic of a free particle and we may expect the same for the energymomentum “coordinates” (E, p). Energy and momentum are conceptually similar: momentum is nothing else than directional energy. Any change in energy should produce a proportional change in momentum (and vice-versa) and this implies a linear energy-momentum relation. Anyway, the best motivation is, perhaps, a curiosity driven one and therefore we ask what are the consequences of postulating a linear energy-momentum relation; they are highly interesting: the existence of spin and antimatter.
III.
GAMMA ALGEBRA
The algebra defined by equation (12) is called a Dirac algebra although it is a special case of a more general one discovered much earlier by W. K. Clifford[3]. It has very interesting features[4]. It is closed with 16 linear independent elements obtained as products of up to four γ µ . All properties of the algebra are consequence of the defining relation, however, it may be convenient to choose an explicit representation of the algebra with 4 × 4 matrices (the smallest possible size). For instance, the Dirac matrices
1 0 , γ0 = 0 −11
γk =
0
σk
−σk 0
built with 2 × 2 identity and null matrices and with Pauli’s matrices 0 1 0 −i 1 0 , σ3 = . σ1 = , σ2 = 1 0 i 0 0 −1
(13)
(14)
If the γ µ are 4 × 4 matrices, then E, p and all variables multiplying them are four component columns. The interpretation of these four elements will be done later. Therefore, the right hand side of equation (12) has an implicit 4 × 4 unit matrix and the left of equation (11) has a unit column. The gamma matrices above are traceless tr(γ µ ) = 0, hermitian or anti-hermitian (γ 0 )† = γ 0 , (γ k )† = −γ k and when squared they result in the identity (γ 0 )2 = 1 , (γ k )2 = −11. The representation given above is not unique. In fact, every transformation with a nonsingular matrix γ µ → U −1 γ µ U also satisfies the defining equation
5 (12). So, besides the matrices given above, called the Dirac basis, there are other choices, for instance, the Weyl basis 0 1 γ0 = , 1 0
γk =
0
σk
−σk 0
,
(15)
or the Majorana basis. An interesting basis, that we will use later, is obtained transforming the “space” part of the Dirac basis with σ3 . This is 1 0 0 σ σ σ 3 k 3 , γk = . γ0 = 0 −11 −σ3 σk σ3 0
(16)
With the algebra of the Pauli matrices one can easily check that these gamma matrices satisfy the Clifford algebra of equation (12).
IV.
LORENTZ TRANSFORMATION
Position xµ and momentum pµ are four-vectors and transform as such under Lorentz transformations xµ → Λµν xν . We will discuss now how the quantities γ µ transform. The notation suggests that they are also four-vectors, however, most books (perhaps all ) dealing with relativistic quantum mechanics or quantum field theory assume that the γ µ behave as scalars and are equal in all reference frames. In these cases the effect of the Lorentz transformation is concentrated in the spinor exclusively. This “spinor picture” is inconvenient for our case without quantization but fortunately there is an alternative “vector picture” where γ µ are indeed fourvectors and transform accordingly[5]. This strategy is not only didactically superior but is a necessity in our case. We assume then that the γ µ are the components of a four-vector in Minkowski space γ µ = (γ 0 , γ ). Notice that a Lorentz transformation preserves the anti-commutator relation (12) and therefore the Dirac algebra of the γ µ is valid in all reference frames. With the vectors xµ , pµ , γ µ we can build six scalars: xµ xµ = τ 2 , pµ pµ = m2 , γ µ γµ = 4 1 , xµ pµ = τ m , γ µ xµ = τ , γ µ pµ = m . (17) The first two are the well known invariants related with proper time and mass. The third follows from the Dirac algebra defined by Eq.(12). The value, τ m, of the
6 Lorentz invariant xµ pµ = tE − xp follows from the first two when calculated in the center of mass frame where x = p = 0. This invariant has units of action and has received little attention in classical physics. However, in quantum mechanics it plays important rˆoles: it can be related with the time-energy and position-momentum indeterminacy relations and involves the non-commutation that leads to the quantization recipe. We will study later the last two relations with more details. Here we just mention that the defining equation for γ µ is quadratic and therefore invariant under a sign change γ µ → −γ µ and this is equivalent to a P T transformation in Minkowski space. There is then a sign ambiguity in the last two relations that we fix assuming mass m and proper time τ to be positive. In the three dimensional euclidean space R 3 , the underling space of nonrelativistic physics, besides the scalars that can be built with vectors we can also build pseudo, or axial, vectors (that remain invariant under space inversion). This is the well known vector product that finds useful applications in angular momentum L = x × p whose components are Lj = εjkl xk pl . These components are the result of the contraction ot the tensor xk pl with the total antisymmetric Levi-Civita symbol εjkl . In the Minkowski four dimensional space of special relativity, we can not build four-vectors as in the vector product of R 3 but the contraction recipe can be generalized and we can build the antisymmetric tensor Lµν = εµν%σ x% pσ in order to formalize relativistic angular momentum. It can be shown[6] that this antisymmetric tensor can be decomposed in two three-vectors: the classical angular momentum L = x × p that corresponds to the L0k components of the tensor, and the dynamic mass moment vector N = E x − p t, associated with the Ljk components. Let us consider, then, all antisymmetric tensors that we can build with our vectors xµ , pµ , γ µ . The tensors εµν%σ x% xσ and εµν%σ p% pσ vanish identically because they are the contraction of a total antisymmetric tensor with a total symmetric one. We already know the angular momentum tensor Lµν = εµν%σ x% pσ . Interesting is the tensor S µν = iεµν%σ γ% γσ that does not vanish identically because γ% γσ is not symmetric (the gammas do not commute). In fact, S µν is an element of the gamma algebra that will become a physical interpretation. The same as happens with angular momentum Lµν , this antisymmetric tensor can be expressed in terms
7 of two three-vectors: S corresponding to the S 0k components and M related with S jk . The vector S is S = (S 01 , S 02 , S 03 ) = (i[γ 2 , γ 3 ], i[γ 3 , γ 1 ], i[γ 1 , γ 2 ]) ,
(18)
and the vector M is M = (S 23 , S 13 , S 12 ) = (i[γ 1 , γ 0 ], i[γ 2 , γ 0 ], i[γ 3 , γ 0 ]) . In the Dirac basis we can explicitly calculate the commutators and we get σ 0 0 σ , M = −2i , S = 2 0 σ σ 0
(19)
(20)
(where σ is a thee-vector with components σk ). The Pauli spin matrices σ completely determine the tensor S µν . Finally, for completeness, we just mention the two tensors εµν%σ γ% xσ and εµν%σ γ% pσ .
V.
PHYSICAL INTERPRETATION OF GAMMA
In order to find an interpretation of the four-vector γ µ = (γ 0 , γ ) we can find out in what direction does the vector γ points in physical space R 3 . In mathematical terms we say that a vector v points in the direction of a unit vector e = (e1 , e2 , e3 ) if the projection of v in e, that is, the scalar product v · e = vk ek , is maximum. Let us build then γ ·e and maximize the resulting determinant. 0 σk ek γ ·e = γ k ek = −σk ek 0
(21)
and its determinant is
2
e3 e1 − ie2 = −e12 − e22 − e32 2 = 1 . det(γγ ·e) = det(σk ek )2 = e1 + ie2 −e3
(22)
This is the same value for all e, that is, the vector γ points everywhere in physical space, a strange vector indeed! This is strongly reminiscent of the result in quantum mechanics where the observation of spin in every direction has the same result (± 21 )
8 and suggests that we associate γ with the spin observable. This suggestion becomes more support when we calculate the vector product γ × γ in physical space R 3 . The j component of this vector, (γγ × γ )j = εjkr γ k γ r is written with (13) in terms of Pauli matrices, and using σk σr = δkr + iεkrs σs can be calculated with some effort; but much easier is to do it with matrix algebra using σ × σ = i 2 σ σ ×σ 0 σ 0 σ −σ 0 × = γ ×γ = σ 0 σ 0 σ ×σ −σ −σ 0 −σ σ 0 −σ 0 1 0 σ = i 2 = i 2 σ σ 0 0 −σ −11 0 −σ 0 1 γ . = i 2 −11 0
(23)
Again, this is reminiscent of the angular momentum vector J in quantum mechanics that satisfies J × J = i~J (this follows from the commutation relation [Jk , Jr ] = i ~ εkrs Js ). Notice also that from (20) we get γ × γ = S and also we can explicitly calculate that [Sk , Sr ] = 4i εkrs Ss .
(24)
This is a most remarkable result because such a commutation relation is taken as the definition of an angular momentum operator in quantum mechanics but we have derived it without making any quantization postulate. This typical quantum mechanic result is a consequence of the liner energy-momentum relation postulate. The Lorentz invariant associated with the gamma four-vector, γ µ γµ = 4 1 , implies that this observable never vanishes, it has the same nonzero value in all Lorentz frames and therefore it can be considered as an intrinsic property of the system like mass or charge. With all this we have enough motivations for the interpretation of γ µ and S µν as an intrinsic angular momentum: spin. Total angular momentum is then formalized by J µν = Lµν + S µν and J = L + S. We have shown that the γ µ are associated with an intrinsic property of the system (spin) coupled with an external one (orbital angular momentum). We will now present another remarkable feature of γ µ also found in quantum mechanics. The linear relation of the invariant γ µ xµ = τ suggests that somehow γ µ is related
9 with some intrinsic velocity. In fact, let us consider the Hamiltonian H = E arising from the linear energy-momentum relation γ0 E − γ p = m. Multiplying this with γ 0 we get E = γ 0γ p + γ 0 m . Now, using this in the second Hamilton equation v k = x˙ k =
∂E ∂pk
we finally obtain v = γ 0γ .
(25)
For a physical interpretation if this intrinsic velocity we can recall that the vector γ points to all directions and never vanishes. Therefore there is an intrinsic propagation in every direction. This is well known in quantum mechanics where the location of a particle is spreading spontaneously. Again we find a “quantum” effect arising from the gamma observable.
VI.
SPIN AND ANTIMATTER
Since the gamma matrices are of size 4 × 4, the dynamical variables E, p are columns with four components that we denote with a Latin index a = 1, 2, 3, 4. That is, Ea , pa . The first remark is that due to (12) all four components satisfy independently the relativistic relation m2 = Ea2 − p2a and therefore we may think of these four components as four relativistic particles, or particles states or particle type. Another important consideration is that, due to the fact that the representations are not unique and there are many choices, the four components are also not unique: if we change the matrices, the multiplying columns also change and therefore if we find, for instance, that one particular property is associated with one particular component, when we change representation the property in question will be shared by several other components. The meaning of the four components are globally distributed among them and not necessarily assigned exclusively to each one. For a particular interpretation of the four components it is convenient to write
10 the equation m = γ µ pµ = γ 0 Ea − γ pa in the Dirac basis in the form m E1 p = − σ 3 m E2 p4 m E p = − 3 + σ 1 . m E4 p2
(26)
These four linear equations are coupled. Subtracting them they become partially decoupled and we obtain E p E p 1 − σ 1 = − 3 − σ 3 . E4 p4 E2 p2
(27)
Both sides are identical except for a minus sign and the exchange of indexes. In this equation we see that the exchange of components (1, 2) ↔ (3, 4) is equivalent to a sign change in energy and momentum, but considering that these are the components of a four-vector, this is equivalent to a space inversion x → −x and time inversion t → −t, that is, the transformations P T . Now, an important result of quantum field theory is that P T C = 1 where C is charge conjugation (the exchange of matter with antimatter) and therefore the P T transformation is equivalent to the C transformation. With all this, we conclude that the components (3, 4) represent the antiparticles of the components (1, 2). The change of the components 1 ↔ 2 and 3 ↔ 4 has no effect in the Dirac basis. The difference of these components appears in the Weyl’s basis where we get the equations m E p = 3 − σ 3 m E4 p4 m E p = 1 + σ 1 . m E2 p2
(28)
In rigour we should use other symbols for the components: the Ea , pa in this equations are not the same as those in the equation (26); they are linear combinations of them. In this Weyl basis the change of the components (1, 2) ↔ (3, 4) is equivalent σ , that is, the opposite spin. Therefore we conclude that the to the change σ → −σ
11 components (1, 2) in this basis represent the same as the components (3, 4) but with the opposite spin. In the Dirac basis, different components correspond to matter and antimatter but spin direction is entangled, that is, the individual components don’t have definite spin direction. On the other side, in the Weyl basis, different components have different spin direction but are not unequivocally associated with matter or antimatter. The question is now whether there is a basis where each component is clearly associated with one choice of matter or antimatter and also spin direction. The answer is “yes” and the corresponding basis is given in (16) called “interesting” before. The equation in this basis is m E p = 1 − σ3σ 3 E2 −p4 m m E3 p1 = − + σ3σ , m E4 −p2
(29)
where we have absorbed the rightmost σ3 matrix in the p components. In these equations we find both effects: the exchange of the components 1 ↔ 3 and 2 ↔ 4 imply changes in all the signs of E and p, that is, changes matter with antimatter, and the exchange of the components 1 ↔ 2 and 3 ↔ 4 corresponds to a change in spin direction. In this basis the four components Ea , pa are associated one to one with the four possibilities: matter or antimatter with spin up or spin down (in any direction) individually. So, if the component a = 1 is matter with spin up, then a = 2 is matter with spin down, a = 3 antimatter spin up and a = 4 antimatter spin down. We can reach the same conclusions about spin and antimatter studying the invariant τ = γ µ xµ = γ 0 ta − γ xa where ta and xa are the coordinates associated with four components in the basis (16). That is, τ t1 x3 = − σ3σ τ t2 −x4 τ t3 x1 = − + σ3 σ . τ t4 −x2
(30)
12 Here, again, the exchange of the components 1 ↔ 3 and 2 ↔ 4 imply changes in all the signs of t and x, that is, changes matter with antimatter, and the exchange of the components 1 ↔ 2 and 3 ↔ 4 corresponds to a change in spin direction. One last comment concerning antimatter is that a change in sign of the fourvectors xµ and pµ , that is, a P T transformation that is equivalent to a C transformation of matter and antimatter, corresponds to sign change m → −m and τ → −τ in the relations τ = γ µ xµ and m = γ µ pµ . This supports the interpretation that antimatter behaves as matter with negative mass and proper-time (that is, it evolves towards the past instead of doing it to the future).
VII.
CONCLUSION
In this work we have seen that the predictions of the existence of spin and antimatter, that are usually assigned to relativistic quantum mechanics, can be attributed exclusively to relativity with the additional postulate of a linear energy-momentum relation. Therefore the quantization postulates are not required for these features of reality well described by a dequantized Dirac equation. Furthermore, we have seen that the spreading of a quantum state can also be explained in this dequantized theory. There is another example where quantum mechanics is assigned the credit for an effect that really comes from relativity. This is the case of the photon. Although its existence was postulated in order to explain the photoelectric effect that was at the origin of quantum mechanics, it can be proved[7] that its existence follows from the analysis of a relativistic massless classical (non-quantum) particle. In particular, we can show that the energy of such a particle must depend on a frequency, that is, E = hν. The frequency ν is related to its spin and the constant h must be determined experimentally. Of course, these results do not diminish the relevance and importance of quantum mechanics that has provided the quantization of energy essential for the description of the microscopic world. However we may speculate whether another additional postulate (like the linear energy-momentum used here) together with special relativity can explain some quantum effects. This would be a relevant contribution
13 bringing some light to the interpretation difficulties faced by quantum mechanics. Actually, all that is needed is a postulate that forces the commutation relations between position and momentum[8]. Another speculation that this work may trigger is related with the anticommutation relation of the gamma observable in equation (12). The right hand side of this equation involves the metric tensor g µν that in general relativity is promoted to a gravitational field and therefore this equation becomes a relation of the spin observable with the gravitational field. This link between the microscopic physics of spin and the macroscopic one of general relativity may be relevant for the unification of these two conflicting domains of physics. I would like to thank ANSES for diminishing financial support and Pablo Sisterna for useful comments.
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14 (2004). [8] A. C. de la Torre “The ubiquitous XP commutator” Eur. J. Phys. 27 225–230 (2006). (or ArXiv quant-ph/0509171).
