Even if tachyons, virtual particles that exceed the speed of light, exist, we cannot use them for superluminal communication.

Hidden variable theories as a form of realism that oppose quantum mechanics suggest that the measurement results of a quantum system influence distant, quantum-entangled quantum systems at superluminal speeds, like telepathy. However, because many physicists respect the causality of relativity, they believe there is no such transmission of superluminal physical influence, and they reject hidden variable theories themselves.

To believe in hidden variable theories and their realism, one must inevitably incorporate non-local elements, and some might think that the "superluminal speed" appearing in that story is realized by elementary particles called tachyons. These are hypothetical elementary particles with purely imaginary mass, and their speed of motion always exceeds the speed of light. If it were argued that these tachyons intervene between distant quantum systems in Bell inequality experiments, would that increase the credibility of hidden variable theories? In this article, let us consider whether these tachyons can truly realize superluminal communication.

This topic is also related to the Higgs field in the Standard Model of particle physics and models of the inflaton field in the early universe. In perturbation theory around points where the second derivative of the field potential V(φ) is negative, instabilities called tachyon modes actually appear, yet physics students often do not know why these do not cause superluminal motion as tachyon particles. I believe the reason can be understood below.

In the following, we adopt a natural unit system where the speed of light and the reduced Planck constant are set to 1.

In relativity, the speed of an ordinary object with positive mass never exceeds the speed of light. Writing this as an equation, it looks like this:

If particles called tachyons exist, this inequality changes as follows:

This means that tachyons have purely imaginary mass. To understand this, let us first recall the equations for the energy and momentum of a particle with positive mass m. They are as follows:

From Equation (4), the following identity is automatically obtained:

It is also known that this relation is invariant under the Lorentz transformation of relativity. While energy and momentum themselves change as vectors under Lorentz transformation, the value of the square of the mass on the left side of Equation (5) does not change under Lorentz transformation. In other words, mass takes the same value in any inertial frame.

Next, let us substitute Equation (3), the condition for tachyons, into Equation (4).

If m remains a real number, the energy and momentum of the tachyon would also become imaginary. Since we want to make tachyons interact with ordinary particles and use them for communication, we must formulate them so that the law of conservation of energy and the law of conservation of momentum hold as a whole. Therefore, the energy and momentum of the tachyon must also be real numbers so that they can be added to the energy and momentum of ordinary particles. Thus, letting μ be a real number, we substitute iμ, a purely imaginary number, for m.

Then, the energy and momentum of the tachyon can become real numbers as follows:

From this equation and equation (5), it can be seen that for tachyons,

is the relationship that holds between μ, energy, and momentum. Based on such arguments, physics posits that the mass of a tachyon is a pure imaginary number.

Now, assuming that these tachyons can interact with ordinary particles, let us concretely consider the possibility of superluminal communication using tachyons. For simplicity, the following discussion will be reduced to 1+1 dimensional spacetime.

First, let us consider ordinary particles. Then, the energy E and momentum p of a particle with mass m satisfy the following relationship.

And the momentum can take values from negative infinity to positive infinity.

The energy of a particle with this momentum p is solved from equation (10) as

but usually, only the positive sign is adopted in equation (12).

Next, as a field corresponding to this particle of mass m, we consider a scalar field that satisfies the Klein-Gordon equation.

The plane wave solution to this equation is

where c.c. on the right-hand side denotes the complex conjugate of the preceding term. This serves as the field mode function corresponding to a particle moving at a speed not exceeding the speed of light.

Let us consider the case of exciting this field, writing information into it, and propagating that information through the field. To do this, we can insert a real external field Q that couples with this field φ into the right-hand side of equation (13).

If the distribution of this Q is changed, the distribution of the excited φ also changes. By measuring the distribution of this φ that has reached a distant location, information about what Q was provided can be obtained. Through this, communication via the field φ that describes ordinary particles can be modeled.

In the equation (15), the general solution is given by the following using the retarded Green's function.

The specific form of this retarded Green's function is given below using the zeroth-order Bessel function.

Here, Θ(x) is the Heaviside step function. If an external field is applied as a delta function at spacetime point A where t=0 and x=0, the propagation status of the field φ can be immediately understood from Equation (17). Figure 1 illustrates the spacetime region where the wave propagates.

Due to relativistic causality, information about the external field is transmitted only within the future light cone of spacetime point A, and superluminal communication using these ordinary fields or particles is, of course, impossible.

Now, let us consider communication using tachyons. In this case, the following relationship holds instead of Equation (10).

In this case, the energy E takes values from negative infinity to positive infinity, and its momentum p is given by

If the sign of the momentum is positive, it is a tachyon traveling in the positive spatial direction, and if it is negative, it is a tachyon traveling in the negative spatial direction.

The corresponding Klein-Gordon equation is as follows.

The plane wave solution to this equation, where energy E is a real number, becomes stable and is given by

This is a right-moving wave solution where momentum p is also positive if E is positive, but the expression with the sign of x inverted in Equation (21) is also a solution to Equation (20), which describes a left-moving wave if E is positive.

Now, let us see if the wave of Equation (21) corresponding to this tachyon can be used for superluminal communication. To do this, we introduce an external field Q that excites this field.

This solution is also given in the same form as Equation (16).

However, because we impose the boundary condition that the tachyon progresses to the right, the following Green's function appears instead of equation (17).

When an external field is applied as a delta function at spacetime point A where t=0 and x=0, the tachyon field φ propagates into the spacetime region shown in Figure 2.

This is clearly moving at a superluminal speed. If one wishes to excite a left-propagating tachyon wave, this can be achieved by using a Green's function where the signs of x and x' are reversed in equation (24). By applying a delta-function-like external field Q at spacetime point B, this left-propagating wave propagates through the spacetime region shown in Figure 3.

So, can we perform superluminal communication using this tachyon excitation? The answer is NO. When we want to communicate some information to a distant location, we decide to do so at a certain time, prepare for it, and then send the signal. We did not want to communicate a signal containing specific information from the time of the Big Bang. However, looking at Figure 2, we can see that superluminal communication requires preparation from the infinite past.

In Figure 4, I have added lines of constant time to the spacetime region in the past where Q in Figure 2 does not exist. This reveals that the distribution of the field φ must already exist at a time in the past before spacetime point A, where we make the decision to communicate.

For example, the excitation in Figure 2 only occurs if we are able to prepare the distribution of φ in the blue region of Figure 4 such that it is consistent with its future. However, unlike the virtual particle tachyon, which has a purely imaginary mass, our bodies are made of ordinary matter with positive mass. It is impossible for us to travel back in time to prepare such communication, that is, to arrange the configuration of the tachyon field. Ultimately, even if a tachyon field exists, communication exceeding the speed of light is impossible for us.

Is it then physically meaningless to consider tachyons? Actually, that is not the case. In quantum field theory that satisfies relativistic causality, tachyons are sometimes introduced to describe the instability of the field.

In equation (21), we only considered stable tachyon plane waves, but in fact, there is another solution to the tachyon field equation in equation (20). When the momentum p has a value in the region of

the region of

Equation (26): Unstable tachyon mode solution

This is an unstable solution that diverges as time passes. When a tachyon is included in quantum field theory, this kind of instability physics appears.

Equation (27): Potential of a field with positive mass

Then, as shown in Figure 5, the field value oscillates around the central value of 0 and stabilizes. However, in the case of a tachyon, the potential becomes

and is inverted as shown in Figure 6.

Then, the field, which was initially at φ=0, begins to roll down the potential due to the influence of a slight external field, and the field value exhibits an instability that diverges. This is the physical effect of a tachyon.

In reality, tachyons have never been observed in experiments, but in fact, approximate tachyons are embedded in the Higgs field of the standard theory of elementary particles. Also, there are models that include this approximate tachyon in the inflaton field, which causes inflation, an exponential expansion that occurred in the early universe. For simplicity, if we model it with a real field, the potential of that Higgs field is given in the form of

Here, the constant g takes a positive value. If we draw this in a figure, it looks like Figure 7.

Around φ=0, the sign of the second derivative of the potential is indeed negative, which is a situation similar to Figure 6. However, as the value of φ increases, the fourth-order term of the field in the second term on the right side of Equation (29) increases, the potential has a lower limit, and the instability disappears as a whole. However, if the field starts moving from φ=0 initially, the amplification of the field that can be approximated by Equation (28) is observed during the early stages of that motion. This can be called the effect of an approximate tachyon.

However, even if we use the Higgs field, superluminal communication is not possible for the reasons stated above. Even if the field was φ=0 in the past and an approximate tachyon mode of the Higgs field was excited by an external field Q at a certain spacetime point A, the information of that external field Q propagates only within the light cone region of A. Since the influence of Q does not reach outside that region, information about Q cannot be communicated at superluminal speeds.

If we speak in terms of the tachyon model in 1+1 dimensional spacetime of Equation (22) considered above, it is described not by the Green's function of Equation (24), but by the following retarded Green's function using modified Bessel functions.

Since the modified Bessel function behaves as follows when its argument diverges:

the value of Equation (30) also diverges as t-t' increases, indicating instability. Figure 8 illustrates the spacetime region where the influence propagates when a delta-function-like external field Q is applied to spacetime point A.

The instability due to the approximate tachyon of the Higgs field also spreads in space only at a speed slower than the speed of light, so superluminal communication is impossible. Ultimately, even if we consider tachyons in this way, we cannot justify a realistic hidden variable theory.