Different Aspects of Spin in Quantum Mechanics and General Relativity

Abstract

In this paper, different aspects of the concept of spin are studied. The most well-established one is, of course, the quantum mechanical aspect: spin is a broken symmetry in the sense that the solutions of the Dirac equation tend to have directional properties that cannot be seen in the equation itself. It has been clear since the early days of quantum mechanics that this has something to do with the indefinite metric in Lorentz geometry, but the mechanism behind this connection is elusive. Although spin is not the same as rotation in the usual sense, there must certainly be a close relationship between these concepts. And, a possible way to investigate this connection is to instead start from the underlying geometry in general relativity. Is there a reason why rotating motion in Lorentz geometry should be more natural than non-rotating motion? In a certain sense, the answer turns out to be yes. But, it is by no means easy to see what this should correspond to in the usual quantum mechanical picture. On the other hand, it seems very unlikely that the similarities should be just coincidental. The interpretation of the author is that this can be a golden opportunity to investigate the interplay between these two theories.

1. Introduction

One of the most fundamental problems in physics today is to find a framework that includes both quantum physics and general relativity. The problem with this unification is not primarily that these theories cover very different areas of physical reality. More essential is that they rely on very different concepts. For example, there have been numerous attempts to treat gravity as a quantum field theory so far without much success, however. On the other hand, we know very little about the true meaning of curvature in quantum physics.

This paper is part of a search for a kind of common kernel for both general relativity and quantum physics. But, in view of the difficulty and complexity of the problem, I have here chosen to work with just one very specific aspect, namely the rotation of elementary particles. There are many reasons for this choice, but to make a long story short, let me just say that there are deep, unsolved problems with this concept in both theories, and in both cases, these involve Lorentz geometry, which clearly must be a necessary part of the unification.

In quantum mechanics, rotation is closely linked to the concept of spin, which can be viewed as a broken symmetry: the equations of motion are invariant under rotations of space, whereas their solutions, in general, are not, but they rather tend to have very specific directional properties.

The algebraic foundation for spinors goes back to a paper of Élie Cartan from 1913 [1]. However, it was through the work of Wolfgang Pauli [2], Paul Dirac [3], and others that its vast importance to physics became clear. Formally, the idea was first introduced to improve Bohr’s model of the atom, but it was soon realized that it could open up new possibilities and new problems. Pauli, in the beginning, opposed the idea that it had something to do with rotation. In fact, for a classical particle to generate the necessary amount of angular momentum, it would have to rotate at such a high speed that it would contradict the special theory of relativity. Nevertheless, it soon became clear that angular momentum must be involved in a fundamental way.

However, it was in connection with Dirac’s attempts to generalize the Schrödinger equation to include special relativity that the real breakthrough came: as it turned out, the scalar field that worked well in Schrödinger’s non-relativistic setting could not be generalized to the relativistic case and had to be replaced by something higher-dimensional. And, the natural choice turned out to be to use the spin formalism, as suggested by Pauli. The theory that emerged can be seen as an elegant way to generalize the Schrödinger equation to include special relativity, but it also became a giant step forwards toward modern quantum field theory.

Was Pauli right to oppose considering spin as a kind of rotation? Or, is it perhaps, in fact, a rotation in a very fundamental sense? What we know from Dirac’s work is that spin is somehow an inevitable consequence of the indefinite metric, but so far, we do not understand why. In quantum mechanics, it is often said that spin is an internal property of elementary particles (with no classical counterpart), but such a statement does not really explain very much: what we have is an extremely well-working formalism, but we may still lack the right conceptual framework. Michael Atiyah suggested that mastering the true meaning of spinors can be seen as learning how to “take the square root of geometry” and that this, in a sense, is analogous to how we strived to understand the meaning of the square root of $-1$, a process that took mathematicians several hundred years [4].

This may very well be so. However, if that is the case, it is important that we learn to ask the right questions. The part of the concept that is usually considered to be the most enigmatic part is the way it behaves under spacial transformations and, in particular, the remarkable fact that a 360-degree rotation does not bring a spinor back to its original state. However, in this paper, it is rather time and the relation to the metric of relativity that is the central enigma. It should also be kept in mind that it has been common wisdom from the early days of quantum mechanics that everything that has to do with gravitation can be neglected in comparison with the other forces since the gravitational force is so weak. And, in particular, curvature has little or nothing to do with quantum mechanical phenomena. In this paper, I want to try a different perspective. Even if it is true that the deviation of the metric of empty space-time from the flat Minkowski metric, caused by the presence of matter, is almost always negligible for problems in particle physics, this does not necessarily mean that the non-zero Ricci curvature (also caused by the presence of matter) must be negligible as well.

This may sound like a rather hairsplitting kind of distinction, but it is part of the ambition of this paper to argue that this should be taken seriously. For the moment, let us note that according to the well-established point of view, every particle with non-zero mass must be surrounded by a non-trivial Schwarzschild metric. And, as was hinted above, the deviation of this metric from the usual one is virtually impossible to observe experimentally. However, close to or inside the particle, something else must be happening since there is no solution to the vacuum equations that extends the Schwarzschild metric to all of space, including the interior of the particle. The perhaps most common way to deal with this problem is to just accept that we have a singularity at the center of the particle and then forget about it. From the point of view of the author, however, this may lead us to overlook something very essential. Another point of view, perhaps closer to reality, is to say that the metric must be well-defined in all of space-time, but that as a consequence, the Ricci curvature cannot be zero everywhere.

This, of course, leads to questions about the nature of this metric, which we are very far from having any methods to answer. And, this is probably also why they are very seldom asked. However, the point of view of the present paper is that there may still be phenomena that we can analyze without any detailed knowledge of the properties of the metric and that we may still have something essential to say about quantum mechanics.

Returning now to the central theme of this paper, the idea that I want to explore can loosely be formulated as follows:

Something in Lorentz geometry makes rotational motion simpler and more natural than non-rotational motion.

As already mentioned above, Lorentz geometry somehow forces spin to enter the wave equations. With respect to general relativity, the idea is instead that this has something to do with the non-zero curvature inside or close to the particle. It is part of the purpose of this paper to clarify this point.

More specifically, what happens when various metrics rotate in space is studied, and my starting point is that curvature can be taken as a measure of complexity. And, as a consequence, optimal simplicity is obtained by minimizing the curvature. It can also be noted that non-zero curvature is linked to the presence of mass energy, and, hence, in a certain sense, minimizing curvature also means minimizing energy or action, although this statement as it stands is an oversimplification.

It is not claimed that this idea of complexity can be directly transferred to quantum mechanics, nor is it claimed that these ideas explain to us the true meaning of spinors or tell us how to take the square root of geometry. In fact, it may very well be that the ultimate description of microcosmos must be discrete and, hence, that all concepts from differential geometry as well as concepts like angular momentum essentially lose their meaning at very short distances and may have to be replaced by something different.

To the mind of the author, no final unification is possible as long as we do not have a theory that can explain to us exactly how matter manages to curve space-time at the micro-level and, in particular, tells us what the metric looks like there. And, it goes without saying that this theory must be covariant in a very fundamental way. But, such a theory may still be far away, so what can be a possible strategy to follow presently?

The choice of the author of this paper has been to start from something that is as close to our classical intuition for rotation as possible and then try to see what effects small modifications of this picture may lead to. In particular, this means considering a setup where the curvature is small, the speed of rotation is low, and the geometry is uncomplicated. The resulting theory may neither be quantum mechanical nor covariant in the usual sense. On the other hand, no particular anomalies occur. When these simplifying conditions are no longer fulfilled, strange phenomena, e.g., closed timelike curves as in the Gödel metric (see [5]), may be possible, and a lot more work is needed before all such cases can be included.

In the present context, choosing different metrics seems to yield results that are remarkably consistent, which suggests that something more fundamental underlies them: it is the belief of the author of this paper that there must be a link between the theorems in this paper and quantum mechanical spin and that understanding the origin of these similarities can be a key step towards understanding the deeper connections between quantum mechanics and general relativity.

As a first step towards an investigation along these lines, I have in this paper tried to put certain technical mathematical results in a broader perspective. The proofs of these theorems, to a large extent, have been carried out using symbolic computer calculations and have already been published for the most part (see [6,7]). The corresponding Mathematica code is also available at [8]. However, for the convenience of the reader, a sketch of the proof of Theorem 2 is included in the Appendix. Also, the necessary computations involving scalar curvature can now be made simpler using the theory of twisted products, and these simplifications are sketched in Section 4. Unfortunately, computer computations are still necessary in other parts of the theorems, but hopefully the simplifications that result from using twisted products make it easier to understand the underlying mechanisms in the future.

2. Rotating Metrics in Euclidian and Lorentz Geometry

In general, rotation is a much more complicated concept in Lorentz geometry than it is in Euclidian geometry, so let us start with the latter case. In fact, here, rotation can be viewed as a rigid motion in the space variables, where the time coordinate only enters as a parameter.

Let g be a positive definite, sufficiently differentiable metric on ${\mathbb{R}}^3$, which we, for simplicity, assume to coincide with the standard Euclidian metric $s=d{x}^2+d{y}^2+d{z}^2$ outside some ball B, centered at the origin.

Remark 1.

This is clearly a simplification, but it appears to be a rather harmless one since the metric is expected to converge rapidly to the Schwarzschild metric (with $R_{ij}=0$).

The metric $g_{\theta }$, obtained by rotating g an angle $\theta$ around a certain axis (which we, without loss of generality, may take to be the z-axis), is then defined in a natural way as the pullback:

$$g_{\theta }=U_{\theta }^{*}g,$$

(1)

where $U_{\theta }$ is the ordinary standard linear rotation operator with matrix

$$M_{\theta }=\left(\begin{array}{ccc}cos\theta & sin\theta & 0\\ -sin\theta & cos\theta & 0\\ 0& 0& 1\end{array}\right).$$

(2)

Clearly, because of the assumption that g equals the standard (rotation invariant) metric outside B, the same is true for the rotated metric. It is also clear that there is a natural way to measure the velocity of the angular rotation (of a non-rotationally invariant metric) as a rotated angle per unit of time.

Given these observations, we can now extend the metric g on ${\mathbb{R}}^3$ to a metric on ${\mathbb{R}}^4$, rotating with angular velocity b, in the following way:

$$\mathfrak{g}\left(b\right)=d{t}^2+{\pi }^{*}{g}_{bt},$$

(3)

where $\pi$ is the canonical projection $\pi (t,x,y,z)=(x,y,z)$. The metrics of this type in this paper turn out to be examples of a common construction in differential geometry, called twisted products; see [9] and Section 4 below.

If we now turn to Lorentz geometry, it is no longer possible to separate space and time when there is motion. A lot of work has been conducted to study rotations in general relativity, for example, to find solutions to Einstein’s vacuum equations $R_{ij}=0$ around rotating objects, notably the Kerr metric [10].

The present paper, however, is not so much concerned with vacuum solutions but rather with metrics with non-zero Ricci curvature. In a sense, this is rather related to the situation where matter is present. However, in this case, there does not even seem to exist a fully satisfactory definition of what we mean by a rotation.

In the following, we partly avoid the problems with the exact meaning of the concept of rotation by mainly studying very slow rotations, where there is an obvious parallel to the Euclidian case. In particular, we may take the metrics $g_t$ on ${\mathbb{R}}^3$ in Section 2 and extend to a metric on ${\mathbb{R}}^4$, but now with the appropriate minus sign for Lorentz geometry:

$$\mathfrak{g}\left(b\right)=-d{t}^2+{\pi }^{*}{g}_{bt}.$$

(4)

To measure the curvature of the metrics in this paper, I have chosen to work with the integral

$$I=\int R^{2}dV,$$

(5)

where R is the four-dimensional scalar curvature and $dV$, in principle, denotes the four-dimensional volume measure. However, since all applications of this measure here are to situations where the amount of curvature is the same for all moments of time, it is simpler and more practical to just integrate over a fixed hypersurface $t=t_0$ (where $t_0$ can be chosen arbitrarily), using the natural three-dimensional volume measure instead. There are many reasons for the choice of I in (5), both mathematical and physical. I briefly return to this question in Section 5.

So, what will happen more specifically if we apply the measure of curvature in (5), as a function of the angular velocity b, to the metrics in (3) and (4)? In other words, let us investigate the function

$$I\left(b\right)={\int }_{{\mathbb{R}}^3}R{\left(\mathfrak{g}\left(b\right)\right)}^{2}dV={\int }_{B}R{\left(\mathfrak{g}\left(b\right)\right)}^{2}dV,$$

(6)

both in the Euclidian case and in the Lorentz case. Clearly, if the three-dimensional metric g itself is rotation invariant, then $I\left(b\right)$ is just constant. But, if we start to compute $I\left(b\right)$ for metrics g that are not rotation invariant, a striking difference between the Euclidian case and the Lorentz case appears. An example is shown in Figure 1.

There are several things that are remarkable about these plots. In the Euclidian case, the most obvious guess is also the best one: the rotation somehow bends the geometry in the time direction, which should create additional curvature. In other words, $I\left(b\right)$ should be minimized when $b=0$ and grow from there in both directions ($I\left(b\right)$ is obviously an even function).

But, in the Lorentz case, the non-rotating metric corresponding to $b=0$ is not minimizing the curvature, not even locally, and starting to rotate it gives rise to metrics with less curvature. This is certainly very much counter to our usual intuition, but it seems to be a very widespread phenomenon. In fact, another remarkable thing about these graphs (which I can not fully illustrate here) is that the shapes of the plots seem to be more or less independent of the exact form of g, except for various scale factors (see also [6] for additional material on this). This should so far be considered as an empirical numerical fact rather than a mathematical one, and it may turn out to be extremely difficult to obtain a mathematically complete picture.

In the following, I, therefore, concentrate on somewhat more restricted statements, which can be proved mathematically and still clearly pinpoint the difference between the Euclidian case and the Lorentz case.

3. Examples of Metrics That Are Not Rotationally Invariant

In Figure 2 and Figure 3, I illustrate two simple cases of twisted products. There are obvious problems with drawing four-dimensional metrics, so in particular, one dimension has been discarded.

Figure 2 depicts a metric that is not rotation invariant but differs from the standard metric only in a region that has a slightly ellipsoidal form. In Figure 3, I instead depict a metric that only differs from the standard one within two spheres. The convex hull of these two spheres forms a region that can be called a string, although, for illustrative purposes, this string has been chosen to be very short and thick. To the right, in both figures, I attempt to add a time dimension to illustrate what happens when these metrics start to rotate around the given axis.

The method of attack is to compute the second derivative $I^{″}\left(0\right)$ for fairly general classes of metrics of the illustrated types, but with the restriction that the deviation from the Minkowski metric is so small that it can be treated as a perturbation. In this case, it turns out that $I^{″}\left(0\right)$ can actually be computed, although the computations become so voluminous that they have had to be carried out on a computer. But, the final results are reasonably simple, and in particular, it can easily be seen that $I^{″}\left(0\right)$ is positive in the Euclidian case and negative in the Lorentz case. The rigorous formulation of these results is the content of the next section.

It should be added that numerical computations show that, in the Lorentz case, when b gets large enough, $I\left(b\right)$ starts to grow again, just as in Figure 1. So, the natural conclusion should be that, for all metrics of the types discussed here, there is an optimal speed of rotation that minimizes the curvature. However, it may be hard to give a rigorous proof of this, and the same goes, in fact, for all statements concerning $I\left(b\right)$ when b is so far from zero that we cannot use the Taylor expansion.

4. Twisted Products

In this Section, I state the rigorous theorems that correspond to the discussion in Section 2 and Section 3. The metrics on ${\mathbb{R}}^4=\mathbb{R}\times {\mathbb{R}}^3$ considered here are twisted metrics of the form

$$\mathfrak{g}=\pm d{t}^2+{\mathsf{\Phi }}^2\left(d{x}^2+d{y}^2+d{z}^2\right),$$

(7)

where $\mathsf{\Phi }(t,x,y,z)$ is a non-zero sufficiently smooth function. Since both factors are standard Euclidian spaces, all the specifics of the product are contained in this function.

Twisted products are well-known objects in differential geometry, so there is a convenient formula for computing the scalar curvature (see [9]). The formula is rather long, but in the present situation, where the factors are trivial and there are obvious frame fields, it simplifies to

$$R=\mp 6\frac{{\partial }^{2}log\left(\mathsf{\Phi }\right)}{\partial t^2}+2{\mathsf{\Phi }}^2\left({\left(\frac{\partial }{\partial x}log\mathsf{\Phi }\right)}^2+{\left(\frac{\partial }{\partial y}log\mathsf{\Phi }\right)}^2+{\left(\frac{\partial }{\partial z}log\mathsf{\Phi }\right)}^2\right)$$

$$-\frac{4}{{\mathsf{\Phi }}^3}\left(\frac{\partial }{\partial x}\left({\mathsf{\Phi }}^2\frac{\partial }{\partial x}log\mathsf{\Phi }\right)+\frac{\partial }{\partial y}\left({\mathsf{\Phi }}^2\frac{\partial }{\partial y}log\mathsf{\Phi }\right)+\frac{\partial }{\partial z}\left({\mathsf{\Phi }}^2\frac{\partial }{\partial z}log\mathsf{\Phi }\right)\right).$$

(8)

We can now formulate the theorems corresponding to the results informally described in Section 2 and Section 3. Starting from a non-zero, sufficiently smooth one-variable function f with compact support in $[0,\infty [$ and $f_{+}^{\prime }\left(0\right)=0$, we can construct a non-rotationally symmetric function by stating

$$h_{\delta }=h_{\delta }(x,y,z)=f(x^2+{(1+\delta )}^2{y}^2+z^2).$$

(9)

Clearly, the level surfaces of $h_{\delta }$ are now slightly ellipsoidal for any small but non-zero $\delta$. If we let

$$h_{\delta ,b}=h_{\delta ,b}(x,y,z,t)=f({(xcosbt-ysinbt)}^2+{(1+\delta )}^2{(xcosbt+ysinbt)}^2+z^2),$$

(10)

then

$$\mathfrak{g}={\mathfrak{g}}_{ϵ,\delta ,b}=\pm d{t}^2+(1+ϵh_{\delta ,b})d{x}^2+(1+ϵh_{\delta ,b})d{y}^2+(1+ϵh_{\delta ,b})d{z}^2,$$

(11)

represents a slowly rotating metric, which is close to the standard metric for small $b,\delta ,ϵ$. Using (8) with $\mathsf{\Phi }=\sqrt{1+ϵh_{\delta ,b}}$, we now have the following theorem:

Theorem 1.

For any sufficiently smooth function f and the metric $\mathfrak{g}$ as in (11) above, $I\left(b\right)$ is a smooth function, and the second derivative at $b=0$ is given by

$$\frac{{\partial }^{2}I}{\partial b^2}\left(0\right)=\pm \frac{1024}{5}\pi {ϵ}^2{\delta }^2{\int }_0^1{t}^{7/2}{\left(f^{″}\left(t\right)\right)}^{2}dt+O\left({({\delta }^2+{ϵ}^2)}^{5/2}\right).$$

(12)

In particular, for any fixed (non-zero) ratio between δ and ϵ and if their magnitudes are sufficiently small, the second derivative is strictly positive in the Euclidian case and strictly negative in the Lorentz case.

As another example of the same phenomenon, we can consider a string (in the sense of Section 3), which we may take to be the convex hull of closed balls $B_{\delta }^{-}\left(t\right)$ and $B_{\delta }^{+}\left(t\right)$ of radius $\delta$, $0<\delta <1$, centered at the points $(-1,0,0)$ and $(1,0,0)$. In the following, for simplicity, let $\delta =1/2$.

Remark 2.

The name “string”, of course, rather refers to the case when δ is a very small number. However, for the mathematics in this section, the exact magnitude of δ is not at all important, and any choice of δ, $0<\delta <1$, works equally well.

If we now let f be a non-zero, sufficiently differentiable one-variable function with support in $[0,1/2]$ and $f_{+}^{\prime }\left(0\right)=f^{\prime }\left(\frac{1}{2}\right)=0$, we get a metric representing a rotating string similar to the one in Figure 3 by letting

$$h_b=f({(x-cos\left(bt\right))}^2+{(y-sin\left(bt\right))}^2+z^2)+f({(x+cos\left(bt\right))}^2+{(y+sin\left(bt\right))}^2+z^2),$$

(13)

and

$$\mathfrak{g}\left(b\right)=\pm d{t}^2+(1+ϵh_b)d{x}^2+(1+ϵh_b)d{y}^2+(1+ϵh_b)d{z}^2.$$

(14)

Theorem 2.

For any f as above and ϵ sufficiently small, the second derivative of the function $I\left(b\right)$ at $b=0$ is given by

$$I^{″}\left(0\right)=\pm 16{ϵ}^2{\int }_{w\le 1/4}{\left(4w{f}^{″}\left(w\right)+6f^{\prime }\left(w\right)\right)}^{2}dV+O\left({ϵ}^3\right),$$

(15)

where $w=x^2+y^2+z^2$. In particular, $I^{″}\left(0\right)$, for sufficiently small ϵ, is strictly positive in the Euclidian case and strictly negative in the Lorentz case.

The proofs of both theorems are very long, even if the computations of the scalar curvatures are now considerably simplified using (8) as compared to the original proofs. The remaining part, involving Taylor expansion and symbolic integration, has, in fact, mainly been carried out with Mathematica on a computer. For Theorem 1, see [6,8]. The proof of Theorem 2 was first published in [7], but it is also included in the Appendix A (see also [8]).

In the theorems in this section, possible changes of the metric in the time direction have essentially been neglected. This is not because such metrics should be considered to be more natural than others in this context; they are just very much easier to handle. However, it should be pointed out that for the main point to be made in this paper, this drawback is not relevant. In fact, the implication that I want to emphasize is that there are metrics on ${\mathbb{R}}^3$ for which the non-rotating (i.e., homogeneous) extension to ${\mathbb{R}}^4$ is not curvature-minimizing among all extensions in the Lorentz case. And, for this, any kind of rotation that gives a negative second derivative will do.

An interesting question is if this also is true for metrics of the type, e.g.,

$$\mathfrak{g}=-{\mathsf{\Psi }}^{2}d{t}^2+{\mathsf{\Phi }}^2\left(d{x}^2+d{y}^2+d{z}^2\right),$$

(16)

where $\mathsf{\Psi }=\mathsf{\Psi }(x,y,z,t)$ is some suitable function. For $\mathsf{\Psi }$ sufficiently close to one in some suitable norm, this should be true for pure reasons of continuity.

For more general choices of $\mathsf{\Psi }$, we can make use of the fact that the metric $\mathfrak{g}$ in (16) is a so-called double-twisted product and again compute the scalar product using the formula in [9]. In this case, however, the analysis is considerably more complicated. I just quote one example here.

In fact, consider metrics of the form

$${\mathfrak{g}}^{\alpha }\left(b\right)=-{(1+ϵh_{\delta ,b})}^{\alpha }d{t}^2+(1+ϵh_{\delta ,b})d{x}^2+(1+ϵh_{\delta ,b})d{y}^2+(1+ϵh_{\delta ,b})d{z}^2,$$

(17)

whereas before

$$h_{\delta ,b}=h_{\delta ,b}(x,y,z,t)=f({(xcosbt-ysinbt)}^2+{(1+\delta )}^2{(xcosbt+ysinbt)}^2+z^2),$$

(18)

for some suitable smooth function f. We now have the following:

Theorem 3.

For any sufficiently smooth function f, where ϵ and δ are sufficiently small, and the metric ${\mathfrak{g}}_{ϵ,\delta ,b}^{\alpha }$ as in (17) above, the second derivative $I\left(b\right)$ at the origin is given by

$$\frac{{\partial }^{2}I}{\partial b^2}\left(0\right)=-(\alpha +2)\frac{512}{5}\pi {ϵ}^2{\delta }^2{\int }_0^1{t}^{7/2}{\left(f^{″}\left(t\right)\right)}^{2}dt+O\left({({ϵ}^2+{\delta }^2)}^{5/2}\right).$$

(19)

Clearly, for $\alpha =0$, this reduces to Theorem 1, and for $\alpha >-2$, the situation is analogous to this case. On the other hand, for $\alpha <-2$, the corresponding argument breaks down, and it is quite possible that, in this case, the non-rotating metrics are also the least curved ones. The proof is similar to the proof of Theorem 1, but see also [8].

However, it is not at all clear what conclusions to draw from this. The metrics in Theorem 3 are not claimed to be particularly natural or representative, and it is not even clear what natural and representative means in this situation. A lot of further research is needed before we can claim to understand these phenomena.

5. Discussion and Conclusions

The theorems in Section 4 have something to say about Lorentz geometry. But, exactly what is it that they want to tell us? Let me again emphasize that none of the metrics in this paper is intended as a realistic model for realistic particles, nor are they constructed starting from a covariant theory, and no such construction seems to be easily within reach at our present level of understanding. The point is instead that even very different metrics seem to give remarkably similar results, which at least suggests that they rest on fundamental principles and, hence, that similar results can also be true for more realistic metrics.

To look for an answer to the above question, it may be appropriate to first ask ourselves what the significance of the quantity

$$I=\int R^{2}dV,$$

(20)

is in four-space? From a mathematical point of view, it is easy to argue that this is the most natural way to measure curvature, at least scalar curvature. In fact, the square norm is, in general, the most natural choice. But, are there any physical arguments? In [11], the author argued that (20) can be viewed as something that is close to a natural Lagrangian for general relativity, with a certain twist, however: it is well-known that there are many Lagrangians that can reproduce the field equations of general relativity in the usual way [12], but (20) is not one of them. In fact, any metric with $R=0$ is stationary for I, but there are lots of such metrics that do not fulfill $R_{ij}=0$. Only when we apply a multiple history perspective are the correct field equations singled out (see [11]).

In other words, the physical motivation for (20) as a Lagrangian is based on another possible link between general relativity and quantum physics. The method used to achieve this uses a kind of “Ensemble approach” to multiple histories, and this actually gives an explanation in the tradition of statistical mechanics for why metrics satisfying the vacuum equations are much more likely than others. But, we are still far from having a sufficient framework for doing an uncontroversial computation of the correct Lagrangian that includes quantum mechanics in a satisfactory way, so clearly, this viewpoint also leads to very difficult questions that cannot be resolved here.

Nevertheless, if we accept (20) as an appropriate Lagrangian, then minimizing this Lagrangian can be viewed as an instance of the Principle of Least Action, and the corresponding integral over a certain hypersurface $t=t_0$ then becomes action per unit of time, i.e., in this case, the rest energy of the particle. So, the answer that general relativity gives to the question of why spinning particles are simpler and more natural is that the rotation itself reduces the restmass of the particle.

It may also be worth pointing out that this view somehow demands that particles, to their very nature, are not rotation-symmetric. An example of a theory that is based on such an assumption is string theory. Perhaps as a matter of curiosity, it can be noted that numerical computations [7] show that the optimal speed for minimizing the curvature of a very thin string (i.e., $\delta$ small), in the sense of this paper, seems to coincide with the situation where the ends of the strings move with, at least approximately, the speed of light, something that is a well-known result about open strings in standard string theory.

But, even if it is very tempting to say that spinning particles are more natural because they correspond to states of lower energy, we are very far from understanding what the actual connection to quantum mechanics is like. Nevertheless, trying to understand the connection between the different aspects of spin in this paper can work as a bridge between the two theories (in both directions).