Hidden Variables with Nonlocal Time

Abstract

To relax the apparent tension between nonlocal hidden variables and relativity, we propose that the observable proper time is not the same quantity as the usual proper-time parameter appearing in local relativistic equations. Instead, the two proper times are related by a nonlocal rescaling parameter proportional to |ψ|², so that they coincide in the classical limit. In this way particle trajectories may obey local relativistic equations of motion in a manner consistent with the appearance of nonlocal quantum correlations. To illustrate the main idea, we first present two simple toy models of local particle trajectories with nonlocal time, which reproduce some nonlocal quantum phenomena. After that, we present a realistic theory with a capacity to reproduce all predictions of quantum theory.

Appendix: Conceptual Issues: Internal Time, External Time, and Nonlocal Time

Even if the mathematics of this paper is straightforward, a conceptually or philosophically inclined reader may have difficulties to understand the novel concept of nonlocal time on an intuitive level. The intention of this Appendix is to help the reader to get a better conceptual understanding of it. For that purpose, we find useful to first explain the concepts of internal and external time.

In classical mechanics, one usually thinks of time as a parameter (t or x ⁰) existing even without particle trajectories or any other mathematical curves in space or spacetime. We refer to time existing without any curve as external time. By contrast, the concept of proper time s is usually viewed as a parameter defined only on a curve in spacetime (usually a particle trajectory). We refer to time defined only on a curve as internal time.

While proper time is typically viewed as an internal time and t is typically viewed as an external time, in this section we want to explain that these views can also be changed without contradicting any physical facts. There is a sense in which t can be viewed as an internal time, and there is also a sense in which proper time can be viewed as an external time.

The proper time s along a curve in spacetime is given by

$$ ds^2 \stackrel{\mathrm{curve}}{=} dX^{\mu}dX_{\mu} .$$

(61)

The right-hand side of (61) is defined only on a curve, so s in (61) is an internal time. But in that sense, it is easy to see that the Newton time t can also be viewed as an internal time. For example, the trajectory X ⁱ(t) of a free nonrelativistic particle satisfies $m \, dX^{i}(t)/dt \stackrel{\mathrm{traj}}{=} p^{i}$ with a constant momentum p ⁱ. This implies

$$ dt^2 \stackrel{\mathrm{traj}}{=} \frac{m}{2E} dX^i dX^i ,$$

(62)

where the summation over the repeated index i=1,2,3 is understood and E=p ⁱ p ⁱ/2m. Clearly, t in (62) is an internal time in the same sense in which s in (61) is an internal time.

Does (62) imply that the Newton time t is really an internal time? One will say no, because (62) is not the definition of t, but only corresponds to a special case (a free particle with a given momentum p ⁱ). On the other hand, (61) is the definition of s, so s is really an internal time. However, definitions are not facts given by nature. Instead, definitions are conventions chosen by humans. Thus, if one defines t to be given by (62), then t is the internal time. Such an internal definition of t is much more restrictive then the external one, but it may be perfectly sensible if one wants to define time as the reading of a physical clock (where the reading of the clock is identified with the position X ⁱ of its needle).

The moral is that t can be defined either as an internal time or an external time. Both definitions are physical, but the external definition is more general. With the external definition of t, the space and time together can be viewed as a 4-dimensional entity. (Such a 4-dimensional view makes sense even without relativity, but relativity further reinforces the relevance of the 4-dimensional view.) By contrast, with the internal definition of t, the world is better viewed as a 3-dimensional entity. Yet, all these different views are nothing but different interpretations of the same physical facts.

Just as t can be interpreted as either internal or external time, the same is valid for s. Even though it is common to view s as an internal time, it can be viewed as an external time as well. The external view means that (61) is not the definition, but only a special case of some more general theory (such as [13] or the theory exposed in the present paper). In the external view one deals with 5 independent parameters s, x ⁰, x ¹, x ², x ³, so one can think of the world as a 5-dimensional entity. Such a view may seem bizarre at first, but actually it is not much more bizarre than the 4-dimensional view of time and space in nonrelativistic physics. Indeed, in the relativistic theory studied in [13], the relativistic-scalar parameter s is very much analogous to the nonrelativistic Newton time t.

The dimensionality of the “world” further increases when one considers more than one particle. The configuration space for n nonrelativistic particles is a 3n-dimensional space. The coordinates for this space are $x^{i}_{a}$, for a=1,…,n. This together with the external time t makes the total of 3n+1 dimensions. Similarly, a covariant formulation of the dynamics of n relativistic particles requires a 4n-dimensional configuration space, with coordinates $x^{\mu}_{a}$. This together with the external proper time s makes the total of 4n+1 dimensions.

With these multidimensional spaces at hand, we can study general coordinate transformations on these spaces. On the (3n+1)-dimensional space, the general coordinate transformation takes the form

$$ \begin{array}{rcl}x^i_a \rightarrow x'^i_a&=&f^i_a(\mathbf{x}_1, \ldots , \mathbf{x}_n,t) ,\\[6pt]t \rightarrow t'&=&f(\mathbf{x}_1, \ldots , \mathbf{x}_n,t) .\end{array}$$

(63)

In particular, (11) is a coordinate transformation (63) with

$$ \begin{array}{rcl}f^i_a(\mathbf{x}_1, \ldots , \mathbf{x}_n,t) &=& x^i_a ,\\[6pt]f(\mathbf{x}_1, \ldots , \mathbf{x}_n,t)&=& \rho(\mathbf{x}_1, \ldots , \mathbf{x}_n)t + \mathrm{const} .\end{array}$$

(64)

Similarly, on the (4n+1)-dimensional space, the general coordinate transformation takes the form

$$ \begin{array}{rcl}x^{\mu}_a \rightarrow x'^{\mu}_a&=&f^{\mu}_a( x_1, \ldots , x_n,s) ,\\[6pt]s \rightarrow s'&=&f(x_1, \ldots , x_n,s) .\end{array}$$

(65)

In particular, (25) is a coordinate transformation (65) with

$$ \begin{array}{rcl}f^{\mu}_a(x_1, \ldots , x_n,s) &=& x^{\mu}_a ,\\[6pt]f(x_1, \ldots , x_n,s) &=& \rho(x_1, \ldots , x_n)s + \mathrm{const} .\end{array}$$

(66)

The physical meaning of general coordinate transformations is well understood in the general theory of relativity, where the coordinate transformations refer to the 4-dimensional spacetime. In particular, a coordinate transformation of the form t→t′=f(x,t) may correspond to a transformation from a nonphysical coordinate time t to a physical time t′. To some extent, the toy models and theories studied in the present paper are similar to the general theory of relativity, in the sense that our transformations (64) and (66) are also coordinate transformations to a physical time, but in more than 4 dimensions. Since our multidimensional spaces involve an n-particle configuration space, these transformations of the time coordinate are naturally viewed as transformations nonlocal on the usual (3+1)-dimensional spacetime. This is what we mean when we say that the physical time-parameters given by (64) and (66) are nonlocal times.

To conclude, time can be viewed either as an internal time or an external time, depending on the context. The external view is more general and requires one dimension more with respect to the internal view. To understand the concept of nonlocal time, it is useful to think of time as a time external to the configuration space. The external view may be an unusual way of thinking when “time” refers to the relativistic scalar time called “proper time”, but is mathematically and physically consistent.