I Introduction and motivation

A possible correspondence between Ricci identities and Dirac equations in the Newman-Penrose formalism

Towards an understanding of gravity induced collapse of the wave-function?





Anushrut Sharma and Tejinder P. Singh

Indian Institute of Technology Bombay, Powai, Mumbai 400076

Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

email: anushrut@iitb.ac.in, tpsingh@tifr.res.in


ABSTRACT

It is well-known that in the Newman-Penrose formalism the Riemann tensor can be expressed as a set of eighteen complex first-order equations, in terms of the twelve spin coefficients, known as Ricci identities. The Ricci tensor herein is determined via the Einstein equations. It is also known that the Dirac equation in a curved spacetime can be written in the Newman-Penrose formalism as a set of four first-order coupled equations for the spinor components of the wave-function. In the present article we suggest that it might be possible to think of the Dirac equations in the N-P formalism as a special case of the Ricci identities, after an appropriate identification of the four Dirac spinor components with four of the spin coefficients, provided torsion is included in the connection, and after a suitable generalization of the energy-momentum tensor. We briefly comment on similarities with the Einstein-Cartan-Sciama-Kibble theory. The motivation for this study is to take some very preliminary steps towards developing a rigorous description of the hypothesis that dynamical collapse of the wave-function during a quantum measurement is caused by gravity.

I Introduction and motivation

One of the possible ways to understand the outcome of a quantum measurement is to propose that there is a dynamical collapse of the wave-function. Some non-linear physical process causes a breakdown of quantum linear superposition, leading to one or the other branches of the superposition being actually realized randomly in a given sampling of the measurement, in accordance with the Born probability rule. A successful phenomenological description of dynamical collapse is the model of Continuous Spontaneous Localization [CSL] wherein the non-relativistic Schrödinger equation is modified to include a non-linear stochastic component Ghirardi et al. (1986), Ghirardi et al. (1990), Pearle (1976), Bassi and Ghirardi (2003). CSL successfully explains what is observed during a quantum measurement, and its predictions of departures from quantum theory are not contradicted by current laboratory tests and astronomical constraints (for a recent review see Bassi et al. (2013)). What is missing though is an infallible understanding of the physical process responsible for the proposed stochastic modification of the Schrödinger equation.

It has been proposed by at least three different researchers, Karolyhazy Karolyhazy (1966), Karolyhazy et al. (1986), Penrose Penrose (1996) and Diosi Diósi (1987) and their collaborators, that such a physical process could be provided by self-gravity. Although their approaches differ in detail, the overall physical nature of the process could roughly be described as follows. A quantum object, while in motion, distorts the geometry of the spacetime around itself, causing it to fluctuate, by virtue of its own gravity. This fluctuating geometry causes loss of phase coherence in the wave-function, and consequent spatial localization of the wave-function, with the effect inevitably becoming more important for objects with higher mass. This has been demonstrated through model calculations by Karolyhazy and collaborators, and similar results have been obtained by Diosi and by Penrose, and by other researchers who have subsequently analyzed their proposal Frenkel (2002), Giulini and Großardt (2011), Giulini and Großardt (2012), Giulini and Großardt (2013), (also see Hu (2014), Colin et al. (2014)). For recent reviews of gravity induced collapse see Gao (2013) and Diósi (2013). For a fascinating different recent proposal as to involvement of gravity in wave-function collapse through a complex metric see Adler (2014).

Here we follow up on a lead provided by some of Karolyhazy’s results, and we suggest some very preliminary ideas as to how one might proceed, from first principles, towards arriving at a fundamental stochastic nonlinear Schrödinger equation which incorporates the effect of self-gravity to explain collapse of the wave-function during measurement and spatial localization of macroscopic objects. Karolyhazy shows that for an elementary particle of mass , the coherence length over which the wave-function of the particle is localized is given by

(1)

For an extended subject of size the localization length is given by

(2)

It is shown in Frenkel (2002) that for a micro-object of linear size Eqn. (1) continues to hold. [For the two expressions coincide]. Thus Eqn. (2) can be taken as the general expression for localization length, and the following important inferences hold:

(3)
(4)
(5)

Eqn. (2) can be written in the following useful form

(6)

where is of the order of the Schwarzschild radius of the object. We now specialize to the theoretically simpler case where , i.e. objects whose physical size is of the order of their Schwarzschild radius, and hence we would be dealing only with objects which are described as black holes, if they are classical objects. Hence Eqn. (6) becomes

(7)

It is clear that in the classical macro-limit the Schwarzschild radius far exceeds the Compton wavelength as one would expect, and this also implies that , and . The macroscopic black hole limit is obtained for large masses such that Schwarzschild radius far exceeds Planck length, and Compton wavelength is much less than Planck length. Conversely the quantum micro-limit is realized if Compton wavelength far exceeds Schwarzschild radius, i.e. , and . The microscopic quantum limit is obtained for small masses such that Schwarzschild radius is far less than Planck length, and Compton wavelength is much larger than Planck length: the concept of black hole is no longer defined in this limit, and one is dealing with a quantum object characterized by its Compton wavelength.

The macro-limit is well described by general relativity and the Einstein equations for spacetime curvature described by the Riemann tensor, accompanied by classical equations of motion for the particle. The micro-limit is described by the Dirac equation for a relativistic particle of mass . In both cases [general relativity and the Dirac equation] the mass acts as a source term. One could well ask as to how the particle knows whether it should obey Einstein equations or the Dirac equation?! In fact there is no information either in Einstein equations or the Dirac equation which would provide an answer to this question. It is only through experience that we know that micro-objects [such as an electron] should be described by the Dirac equation and macro-objects [such as a solar mass rotating black hole] should be described by Kerr solution of Einstein equations. And then there is the argument presented above, for gravity induced quantum-classical transition, where the behaviour of an object is quantum or classical, depending on whether or not the associated Compton wavelength exceeds the Schwarzschild radius. It would be significant if there were to be a system of equations describing the dynamics of a mass , irrespective of whether or not this mass satisfies or , and to which system the Einstein equations and the Dirac equation would be approximations. Such a system of equations would significantly help in understanding the transition region as well as provide an improved quantitative understanding of gravity induced quantum collapse and localization. A small step in this direction will be proposed in the following sections.

The possibility of such a structure arising is also suggested by schematically looking at a possible action for the gravitational field and the Dirac field, if both were to be sourced by the same mass . Ignoring technical and related conceptual issues, since this is largely a heuristic argument, one could write the action as

(8)

[If one were to imagine getting the Einstein equations out of this action, in the limit , then in the last term one could replace by a spatial three-delta function representing localization of the mass at a point].

Let us estimate the relative magnitudes of the integrands in these three terms in the action, by introducing characteristic lengths, leaving aside the four volume and metric which are common to all three terms. If there is a characteristic length associated with the system, the curvature scalar may be estimated via and the first term is , whereas the second term is and the third term is . If dominates over , then by virtue of the resulting field equations we expect and hence

(9)

If dominates over and is order then

(10)

We see that suggests (the scale implied by should be ignorable) and suggests (the scale implied by should be ignorable). This indicates that when and a dynamical description arising out of a combined consideration of the Dirac action and the Einstein-Hilbert action for a particle of mass might be possible.

It is interesting that the three terms all come with different power-law dependence on the characteristic length scale, being respectively. As a result, we can write the three integrands together in the following form, after pulling out the constant :

(11)

This permits thinking of the Dirac field as a modification arising if Planck length is taken as non-vanishing.

In order to make preliminary progress towards describing Einstein equations and Dirac equations as part of a common system, we must first try and find a common language for describing gravity and the Dirac field. Of course gravity described via the metric and the Dirac field described by spinors look very different. On the other hand, more promising is a consideration of Einstein gravity in the tetrad formalism; in particular through the spin coefficents which are used to write down the Riemann tensor in the Newman-Penrose formalism, via the so-called Ricci identities. As for the Dirac equation, this too can be written for a curved space in the Newman-Penrose formalism, and then the set of four Dirac equations look strikingly similar to the Riemann tensor equations [the Ricci identities] written in term of spin coefficients in the N-P language. We then make the unusual proposal that the Dirac spinor components be identified with four of the spin coefficients. This allows the four Dirac equations to be deduced from the eighteen complex N-P equations for the Riemann tensor, provided one includes torsion in the theory. This brings in some similarity of the present analysis with the Einstein-Cartan theory, which generalizes general relativity to include torsion. Also, it appears necessary to assume a new form for the energy-momentum tensor, which relates to the Ricci tensor and Weyl tensor via a complex generaization of Einstein equations. Since the Dirac equations are in this manner found to be a special case of the Newman-Penrose equations, this raises the hope that a Dirac equation which incorporates the effect of self-gravity (and whose non-relativistic limit might be of help in understanding gravity induced collapse) might be obtainable from the Ricci identities with torsion, written in the N-P formalism.

Ii Einstein equations in the Newman-Penrose formalism

The Newman-Penrose formalism is a tetrad formalism in which the basis vectors are a tetrad of null vectors with and real, and and complex conjugates of each other. These basis vectors can be considered as directional derivatives and are denoted by the special symbols

(12)

The Ricci rotation coefficients, also known as spin coefficients, arise in the definition of the covariant derivatives of the four null vectors. There are twelve spin coefficients, denoted by special standard symbols

(13)

The ten independent components of the Weyl tensor are expressed by five complex Weyl scalars, denoted as

(14)

The ten components of the Ricci tensor are defined in terms of four real scalars and three complex scalars

(15)

The definitions of these scalars can be found for instance in Chandrasekhar’s book The Mathematical Theory of Black Holes Chandrasekhar (1983).

The Riemann tensor can be expressed in terms of Weyl scalars and Ricci scalars, and directional derivatives of the spin coefficients. There are eighteen complex equations to this effect, known as Ricci identities, not all independent. These central equations of the N-P formalism are given by Equations 310 (a-r) of Chapter 1 of Chandrasekhar’s book, and we reproduce them below, for our further use.

(16a)
(16b)
(16c)
(16d)
(16e)
(16f)
(16g)
(16h)
(16i)
(16j)
(16k)
(16l)
(16m)
(16n)
(16o)
(16p)
(16q)
(16r)

In these equations the Ricci components are to be determined from the Einstein equations. Since there are thritysix real equations, and only twenty independent Riemann components, these equations are subject to sixteen constraints, the so-called eliminant conditions.

Iii A proposal for obtaining Dirac equations as a special case of Ricci identities

The four Dirac equations in the N-P formalism are given by [Chandrasekhar, p. 543] Chandrasekhar (1983)

(17)
(18)
(19)
(20)

where and are Dirac spinor components.

Here, . The similarity between the formal structure of the Dirac equations and the Ricci identities is suggestive. In both cases, each equation involves a pair of derivatives of spin-coefficients / Dirac spinors. We will see how under certain assumptions the Ricci identities in the N-P formalism, after a suitable generalization, might contain within themselves, the Dirac equations as a special case. The first generalization we refer to is that the Riemann tensor should be allowed to be complex, as a consequence of the complexity of the spin coefficients [unlike in the standard case where the Riemann tensor is real, even though the spin coefficients are complex]. This complexification of the Riemann tensor does not prevent the assumption of a real flat Minkowski space-time background, because as we will eventually see, the complex components are assumed to arise from a complex non-zero torsion. The second generalization will consist of a new relation between the Riemann tensor and the energy-momentum tensor, which amounts to suggesting a generalized form for the energy-momentum tensor. The principal assumption is that we propose to make a correspondence between the four Dirac spinor components in the N-P formalism, , with four of the spin coefficients. Since a spinor component has length dimensions and a spin coefficient has length dimensions , they will be related through a constant of length dimension which we assume to be the square-root of Planck length . Upon inspection and comparison it is found that the appropriate identification between the spin coefficients and the Dirac spinor components is

(21)

The remaining eight spin coefficients are set to zero:

(22)

The reason why we chose, and , to be the Dirac spinors is that these are the only four spin coefficients which are absent from the Dirac equations. Further, when we take and to be the last two Dirac spinors, we get all the appropriate derivatives from the Ricci identities, as are required by the Dirac equation.

There remains an important unanswered question as to how one could be sure that tetrad transformations will not mix up the spin coefficents and make non-zero those spin-coefficients which we have set to zero? We do not have an answer to this at the present stage, and it remains an assumption. However an indicator that our assumption that these four non-zero spin-coefficients might not mix with the other ones comes from an observation relating to the opposite extreme limit, namely black holes in general relativity: we quote from Chandrasekhar [Chapter 1, Section 9, page 63] Chandrasekhar (1983). “It is a remarkable fact that the black-hole solutions of general relativity are all of Petrov type D and therefore, enable their analysis in a null tetrad frame in which the spin coefficients , and and all the Weyl scalars, except , vanish.” Remarkably, it is precisely these spin coefficients which are required to be non-vanishing in the microscopic Dirac limit, while the other eight are set to zero. It is as if there is some complementarity between the black hole limit and the Dirac limit; this gives us some confidence in the assumptions made here. Furthermore, in the context of the torsion dominated case considered below, it is to be noted that on a Minkowski flat spacetime, the connection is antisymmetric and has four independent components O’Connor and Smrz (1978).

We now examine the eighteen N-P equations to see how the four Dirac equations can be obtained from them. Thus we re-write the Ricci identities under the conditions (21) and (22) :

(23a)
(23b)
(23c)
(23d)
(23e)
(23f)
(23g)
(23h)
(23i)
(23j)
(23k)
(23l)
(23m)
(23n)
(23o)
(23p)
(23q)
(23r)

These equations give the following solutions for the Ricci and Weyl tensors in terms of the spin coefficients

(24a)
(24b)
(24c)
(24d)
(24e)
(24f)
(24g)
(24h)
(24i)
(24j)
(24k)
(24l)

Now we try to obtain the four Dirac equations in terms of these eighteen equations (23). For the first Dirac equation, we add (23g) and (23k), to get

(25)

Let us compare this with the first of the Dirac equations (17) which can be written as

(26)

Using the correspondence conditions (21) and (22) this equation can be written as

(27)

From (25) and (27), we get

(28)

For the second Dirac equation, we add (23m) and (23p), to get

(29)

We compare this equation with the second Dirac equation (18) which can be written as

(30)

Under (21) and (22), this gives

(31)

From (29) and (31), we get

(32)

For the third Dirac equation, we add (23i) and (23a), to get

(33)

We compare this equation with the third Dirac equation (19) which can be written as

(34)

Under (21) and (22), this gives

(35)

From (33) and (35), we get

(36)

For the fourth Dirac equation we add (23n) and (23c), to get

(37)

We compare this equation with the fourth Dirac equation (20) which can be written as

(38)

Under (21) and (22), this gives

(39)

From (37) and (39), we get

(40)

In summary, the N-P equations (25), (29), (33) and (37) reduce to the four Dirac equations (27), (31), (35) and (39) [which are identical to the Dirac equations (17), (18), (19) and (20)] under the correspondence (21) and subject to the four conditions (28), (32), (36) and (40). From the four equations, (28), (32), (36) and (40), we get

(41a)
(41b)

iii.1 Constraints on the spin coefficients

Unfortunately this construction runs into problems. As we know, four of the Ricci components listed in Eqn. (15) are real. These reality conditions must be respected by the solutions (24), and furthermore the eliminant conditions are to be imposed on the Ricci identities since there are only twenty independent Riemann components. Upon examination we find that this imposes the following undesirable constraints on the four Dirac spinors:

(42a)
(42b)
(42c)
(42d)
(42e)
(42f)
(42g)

This gives nine constraints on the four non-zero spin coefficients. These constraints are unacceptable because the four Dirac spinors are in general completely independent of each other, apart from the normalization constraint, and any other constraint which might arise from the intrinsic structure of the Dirac equation.

The origin of these constraints lies in the fact that there are eighteen complex [equivalently thirty-six real] Ricci identities, whereas there are only twenty independent components to the Riemann tensor (See for instance the discussion in Chandrasekhar (1983) Chapter 1.) Now it is known that if one were to include torsion in the theory, the Riemann tensor has thirty-six independent components, and the above mentioned constraints do not arise. This is what we do next: write down the eighteen Ricci identities in the N-P formalism for a Riemann-Cartan space-time which includes torsion. We now find that it is possible to establish a correspondence with Dirac equations without having undesirable constraints.

With hindsight, it becomes clear why this correspondence would not work without including some additional structure, such as torsion. The spin coefficients describe the gravitational force, whose role is assumed to diminish in the microscopic limit, in which limit quantum effects are assumed to become more important. It would be unreasonable to expect these spin coefficients to then take over the role of the Dirac field as the micro-limit is approached. Thus the above costruction should be regarded as a toy model, which does not work, but which makes it easier to understand the following more intricate construction which includes torsion.

Iv Ricci identities for space-time with torsion

We next consider the Ricci identities in the NP formalism with torsion, since this appears to be a possible way for eliminating the constraints which appear on the spin coefficients and consequently on the Dirac spinors. These identities have been worked out for instance by Jogia and Griffiths Jogia and Griffiths (1980). In this case, the connection is given by

(43)

where is the Christoffel symbol of the second kind, and is the contortion tensor which describes the presence of torsion. In this case, the Ricci tensor is no longer necessarily symmetric, and there are six additional components in the Ricci tensor, described by the three complex quantities ( and ). The Weyl tensor has ten additional components, described by the real quantities () and the complex quantities (). For definition of these notations please see Jogia and Griffiths (1980).

Moreover, the spin coefficients now have an additional term due to torsion, which is denoted by . Thus, the spin coefficients can now be written as , where the first term corresponds to the torsion free part and the second term corresponds to the torsion component of the spin coefficients. We further use the following notation to represent the spin-coefficients (see Jogia and Griffiths (1980) for details):

(44)

The Ricci identities are modified when torsion is included, and are now given by Jogia and Griffiths (1980)

(45a)
(45b)
(45c)
(45d)
(45e)
(45f)
(45g)
(45h)
(45i)
(45j)
(45k)
(45l)
(45m)
(45n)
(45o)
(45p)
(45q)
(45r)

These generalize the Ricci identities given previously in the torsion-free case, and as may be verified, reduce to those equations when torsion is set to zero.

Once again, we use (22) to retain only four non-zero spin-coefficients. Thus, equation (45) gives

(46a)
(46b)
(46c)
(46d)
(46e)
(46f)
(46g)
(46h)
(46i)
(46j)
(46k)
(46l)
(46m)
(46n)
(46o)
(46p)
(46q)
(46r)

Solving these equations for the Ricci and the Weyl tensors in terms of the spin coefficients gives the following results

(47a)
(47b)