A Case for the Application of Analytic Continuation in Physics

Abstract

This chapter will motivate the introduction of one proposed physical principle of vacuum diagram loop divergence treatment by analytic continuation in conjunction with the proposed inherent physical property of virtual quantization. And show that when such a proposed principle and proposed property are adopted that it becomes possible to associate a finite negative zero-point energy. This proposed physical principle is shown to be a useful or practical and mathematically equivalent way of interpreting the Casimir effect. Some examples of the application of this proposed physical principle are outlined, in particular its application to the conformally flat de-Sitter case. An outline of the quantitative implications of this model is tabled for the sake of clarity and completeness. Important discussions on the most critical sources of error and falsification are mentioned, and concrete predictions made.

1. Introduction

The quest for the unifying principles of physics started with the work of many brilliant minds more than 50 years ago as of the time of writing of this chapter. In 1973, it was shown [1] by Bekenstein that for the case of black holes, where the principles of physics meet, that it is natural and necessary to consider that its entropy be proportional to the surface area of the black hole and not its volume. Less than a year later, in 1974, Hawking [2] found a way to show that black holes are not perfectly black and must radiate thermally due to quantum effects. The consequence of their work, (and others) was that due to the fact that the internal energy could only originate from the black hole mass and not from other known sources of energy that black holes must evaporate with time and eventually disappear completely, and thus that all the quantum information that falls in would eventually be lost to our universe. More precisely, as recently reviewed by Calmet and Hsu [3] that black holes cause pure states to evolve into mixed states. This paradox came to be known as the information paradox. Approximately at the same time, as an attempt at finding these unifying principles, it was shown by Davies [4] and independently by Unruh [5] that any uniformly accelerated observer in a vacuum field will experience a thermal bath. The equivalence principle provides the bridge between this Unruh-Davies effect and the Hawking radiation at a black hole’s surface.

Ever since the seventies and eighties, there have been many other interesting proposals and developments as for the fundamental, unifying principles, and consequent theories of physics. One of the most promising attempts at this was the development of string theory, built on the work by Veneziano [6], and many others. The use of analytic continuation as a way to treat divergences is often used in string theory, for example, as demonstrated by D’Hoker and Phong [7]. As far as the author of this chapter understands, the issue of the arising vacuum diagram loop divergences in physical calculations is a remaining problem today. This motivates an attempt at introducing a unifying physical principle of how to treat these vacuum diagram loop divergences in general.

In this context, a special topic of interest is the case of virtual particles. As reviewed by Lamoreaux [8], the Casimir effect or Casimir force is known to be an attractive force between two uncharged material bodies (usually, thin parallel plates) due to the difference of the zero-point energies associated with said virtual particles on either side of said plates. This energy gradient generates a small and attractive force between the plates. This interpretation of the origins of the Casimir force originates from the treatment of the arising divergence by zeta regularization and consequent renormalization, a technique for subtracting away the divergence on either side of the plates, in order to be able to associate a finite, physically measured value by Mohideen and Anushree [9] and many others.

With these in mind, this chapter will begin by discussing the treatment of virtual divergences with regards to this effect. And how a different, mathematically equivalent interpretation may be more useful or practical as it comes to the general treatment of these vacuum diagram loop divergences by a proposed, unifying physical principle. This is important as a tool in order to develop an intuition of how this interpretation may be more generally applied. This discussion will later be expanded upon in order to include this plausible general interpretation, and how it applies to an example. The chapter will, then, be expanded further by a different example of the application of this proposed general principle, based on prior work [10] by the author of this chapter. The known immediate implications will be tabled for the sake of clarity and completeness and followed by a section (see Section 3) that will try to address and discuss some of the main issues of the proposed model, as well as offer possible methods of falsification. Finally, concrete predictions based on this proposed general principle of the treatment of vacuum diagram loop divergences will be made. In addition to some concrete predictions based on its implications as shown.

2. A brief outline of the thought process

In the underlying QFTs (see Abbreviations), the processes of regularization refer to the set of techniques known to obtain a finite result of divergent series or loop integrals. For an extensive overview of the various regularization methods see [11] by Kachelriess. What all these methods have in common is that they all try to get rid of arising divergences in various ways. This is part of the motivation of trying to find a unifying principle of how to treat arising vacuum diagram loop divergences in general. One attempt, it is possible to try is inspired by the method of zeta regularization. That is treating the arising vacuum diagram loop divergences by going beyond them instead of getting rid of them; in this way the introduction of regulators or cutoff parameters becomes redundant, and the arising divergences become treatable by analytic continuation. Next follows a discussion of this with regards to the Casimir effect.

2.1 Divergence treatment and the Casimir effect

The vacuum energy density $\rho \left(d\right)$ associated with virtual fluctuations per polarization mode in the 1 + 1 dimensional Casimir case has been shown by Kachelriess [12] and many others to lead to an equation of the form.

$$\rho \left(d\right)=\frac{\pi }{2d^2}\sum _{n=1}^{\infty }n$$E1

where $d$ is the distance between the plates. In this case, the necessary virtual quantization

$${\omega }_n=\frac{\mathit{n\pi }}{d}$$E2

follows from the boundary conditions of the wavefunction between the plates. The divergence of (Eq. (1)) is then treated by zeta regularization and renormalization, see [12] for details. After this divergence treatment, the final result is of the form.

$${\rho }_{\mathit{\text{ζreg}}}\left(d\right)=-\frac{\pi }{24d^2}$$E3

And thus it is possible to obtain an energy distance gradient, and therefore an attractive Casimir force between the plates. This zero-point energy gradient is, thereby, interpreted to be the origins of the Casimir force.

Another useful or practical way to interpret the origins of the Casimir force in order to build a tentative intuition of the general divergence treatment in QFTs is by the application of the principle of vacuum diagram loop divergence treatment by analytic continuation and using the well-known result.

$$\sum _{n=1}^{\infty }n=\zeta \left(-1\right)=-\frac{1}{12}$$E4

The Riemann zeta function $\zeta \left(s\right)$ evaluated by analytic continuation at $s=-1$. It is noted that this is the identical sum as it appears in (Eq. (1)). Thus, by interpreting the treatment of the virtual divergence by analytic continuation (Eq. (4)), it becomes possible to avoid the introduction of a cutoff function and go beyond the divergence instead of trying to get rid of it as follows

$${\rho }_{\zeta }\left(d\right)=\frac{\pi }{2d^2}\sum _{n=1}^{\infty }n=\frac{\pi }{2d^2}\left(-\frac{1}{12}\right)=-\frac{\pi }{24d^2}={\rho }_{\mathit{\text{ζreg}}}\left(d\right)$$E5

which is mathematically equivalent to the regularization treatment and interpretation, which it must be for consistency.

Therefore, according to this interpretation, the Casimir force does not originate from the zero-point energy gradient across the plates but rather originates from an inherent proposed physical principle applied to the virtual fluctuations in the space itself, which then exists independently of the presence of the finite Casimir cavity. The inherent proposed physical principle is the principle of the treatment of the vacuum diagram loop divergences by analytic continuation. In other words, the effect is inherently present and can only be changed in magnitude by finite cavity manipulations.

2.2 General discussions on the proposed inherent physical principle of vacuum diagram loop divergence treatment by analytic continuation

When describing the motion of a free scalar field $\varphi \left(x_i\right)$ in free field theory, a function of interest is often the Green’s function. The reason for this is because the Green’s function can be used as a mathematical tool to solve difficult EOMs (see Nomenclature). In said free field theory, a typical general calculation usually ends up with trying to derive the $n$-point Green’s function $G^{\left(n\right)}\left(x_1,\dots ,x_n\right)$ described by an infinite sum over all loop orders, the first loop order contribution is (see Figure 1).

Figure 1.

The loop diagram of interest to first order with an external line.

Where the vacuum diagram loop propagation without external lines is described by the Euclidean Feynman loop propagator after Wick rotation.

$$i{\mathrm{\Delta }}_F\left(0\right)=\frac{1}{{\left(2\pi \right)}^4}{\int }_{-\infty }^{\infty }\frac{d^4{k}_E}{k_E^2+m^2}$$E6

where it was reiterated in [10] that in order to obtain the desired Weyl invariance it is necessary that $m=0$. Thus, the divergent Feynman loop integral simplifies to

$$i{\mathrm{\Delta }}_F\left(0\right)=\frac{1}{{\left(2\pi \right)}^4}{\int }_0^{\infty }\frac{{\mathit{dk}}_{E}2{\pi }^2{k}_E^3}{k_E^2}=\frac{1}{8{\pi }^2}{\int }_0^{\infty }{\mathit{dk}}_E{k}_E$$E7

which is the divergent vacuum diagram loop integral of interest. It is noted that due to the fact that the discussion is concerning periodic loop diagrams, and thus that there may be a standard periodicity of the field $\varphi \left(x_i\right)$

$$\varphi \left(x_i+L\right)=\varphi \left(x\right)\to e^{{\mathit{ik}}_E\left(x_i+L\right)}=e^{{\mathit{ik}}_E{x}_i}$$E8

which implies

$$e^{{\mathit{ik}}_{E}L}=1=e^{i2\mathit{\pi n}}\to k_E=\frac{2\mathit{\pi n}}{L},n\in \mathrm{\mathbb{N}}$$E9

For the $n$-th mode. Thus, it can be argued that virtual particles which are by definition associated with closed-loop diagrams, and therefore must have periodic boundary conditions with some periodic lengthscale $L$ independent of the spacetime background they are embedded in necessarily have to have a quantized wavenumber as shown.

Therefore, it is plausible that the origins of the quantization are inherent to the virtual particle system by definition, and thus may exist independently of the presence of a finite Casimir cavity. Using this proposed physical inherent property and the proposed general principle of vacuum diagram loop divergence treatment by analytical continuation, it becomes possible to conveniently avoid the introduction of a regulator and instead go beyond the divergence to obtain a result for the vacuum diagram loop integral of interest (Eq. (7))

$$i{\mathrm{\Delta }}_F\left(0\right)=\frac{1}{8{\pi }^2}{\int }_0^{\infty }{\mathit{dk}}_E{k}_E=\frac{1}{4\mathit{\pi L}}\sum _{n=0}^{\infty }n=\frac{1}{4\mathit{\pi L}}\left(-\frac{1}{12}\right)=-\frac{1}{48\mathit{\pi L}}$$E10

For some periodic lengthscale $L$. Thus, it becomes possible to associate a finite and negative zero-point energy contribution in flat Euclidean geometry. It was shown [10] that the application of the proposed physical inherent property of virtual quantization (which may or may not be true for all possible spacetimes) together with the proposed principle of vacuum diagram loop divergence treatment by analytic continuation can with internal consistency be extended to the case of a conformally flat dS spacetime as observed by any inertial observer. This application and subsequent immediate implications of this will be discussed next.

2.3 The application of the general principle of vacuum diagram loop divergence treatment by analytic continuation to the conformally flat dS case

The proposed general principle of vacuum diagram loop divergence treatment by analytic continuation in conjunction with the proposed plausible property of inherent virtual quantization was shown in detail [10] with internal consistency to lead to a zero-point energy $E_0$ within an implicit unit space $L=1$ m, see Section 3.1 for an error discussion on this to be.

$$E_0=-\frac{\pi }{6}<0$$E11

Converting to SI units and making the critical observation of the presence of a $\mathrm{\hslash }c$ factor, which may account for the scaling required in the relevant conformally flat dS case, and finally squaring for dimensional analysis.

$\left[{\mathrm{\Lambda }}_{\mathit{dS}}\right]$$=$ m ${}^{-2}$ yields the critical scaling

$$\boxed{{\mathrm{\Lambda }}_{\mathit{dS}}\propto {\mathrm{\hslash }}^2{c}^2}$$E12

Where the exact value of the geometrical proportionality constant remains to be determined with accuracy but is expected to be of order $O\left(-1\right)$. The critical observation that the proposed principle of vacuum diagram loop divergence treatment by analytic continuation in the conformally flat dS case and indeed for the Minkowski case, as well as shown in the previous Section 2.2 leads to the desired 52 zeros in SI units within an order of magnitude compared to the Planck data [13] was the original motivation for the proposition of this physical principle. For a more detailed derivation of this see [10].

2.4 Immediate implications of the principle of vacuum diagram loop divergence treatment by analytic continuation applied to the conformally flat dS case

It is plausible to argue that the simplest possible hypothesis is that the conformally flat dS cosmology attempted to be described here is identical to an internal Schwarzschild geometry (see Appendix A and [10] for quick derivations with regards to this). All the immediate implications are tabled (see Table 1) for the sake of clarity and completeness. A final less immediate interpretation of another implication is briefly outlined in [16], interpreting the Hubble scale as identical to a quantum uncertainty in position, and therefore may provide a mechanism for why there is something rather than nothing. It is quite possible there are even more unknown implications here, and more work is needed to outline them if they exist.

Quantitative relation in SI units	Name or context	Physical interpretation
${\mathrm{\Lambda }}_{\mathit{dS}}\propto {\mathrm{\hslash }}^2{c}^2$	Lagrange multiplier.	Zero-point energy squared in dS spacetime.
${\rho }_{\mathit{dS}}\propto \frac{{\mathrm{\hslash }}^2{c}^6}{G}$	Ideal fluid model.	Default vacuum energy density in dS spacetime.
$R\propto \frac{1}{\mathrm{\hslash }c}$	Spatial observable cosmological scale today.	Current Hubble scale, that is. the radius to the current cosmological event horizon.
$M\propto \frac{c}{\mathrm{\hslash }G}$	Massive observable cosmological scale today.	Baryonic matter (or effective matter, that is including the dark matter) of the observable cosmological scale today.
$a_{\mathit{dS}}\propto \mathrm{\hslash }{c}^3$	Dark energy.	Repulsive acceleration due to virtual vacuum fluctuations in dS spacetime.
$v^4\propto \mathrm{\hslash }{\mathit{Gc}}^{3}M$	Tully-Fisher relation [14]	Galaxy mechanics.
$w=-1<-\frac{1}{3}$	Black hole singularity theorems.	The presence of an EOS$w=-1<-\frac{1}{3}$counteracts the gravitational collapse [15], and thus resolves the black hole physical singularity.
$M^2\propto {\left(\frac{c}{\mathrm{\hslash }G}\right)}^2$	Plausible baryon mass symmetry breaking mechanism.	$M=0$is not a global minima of the energy function (Eq. (13)), but it rather has two global minima. Presumably corresponding to the matter vs. antimatter solutions.
$\mathit{dS}=-\frac{p_{\mathit{dS}}\mathit{dV}}{T}>0$	Information paradox.	The cosmological entropy is encoded at its surface, the presence of another energy source suggests that the internal energy need not originate from the black hole mass$M$, and thus does not need to evaporate with time, but expands with time at all times.

Table 1.

Table of relations, context, and physical interpretations of said proposed model.

3. Error discussion and methods of falsification

This section will discuss plausible errors of this analysis and end with mentioning possible methods of falsification. The error discussion will focus on two main critical points: the arbitrary chosen $L=1$ m value and the issue of covariance.

3.1 L = 1 m WLOG

One objection made is the issue of the arbitrary chosen $L=1$ m value, and how this can have anything to do with the general cosmological scale. It is of course true that for the proposed model and principle to be physical, that its empirically correlated output must be invariant under our arbitrary chosen size of unit of measurement. As far as the author of this chapter understands as of its time of writing, it is thought that the role of the periodic $L$ lengthscale, in addition to being a measure of the cavity dimension if such a cavity is present at all as outlined in Section 2.1 is that it $\mathit{\text{defines}}\mathit{\text{the}}\mathit{\text{unit}}\mathit{\text{of}}\mathit{\text{measurement}}$ used to derive the quantitative output (see Section 2.4). In other words, setting $L=1$ m and only in this way, allows for the usage of the SI units measured values of the factor $\mathrm{\hslash }c$. Setting $L$ to an arbitrary value say $\alpha$, such that $L=\alpha$ scales the unit of measurement by a factor $\alpha$ such that the then numerically measured value of $\mathrm{\hslash }c$ changes by the same factor as compared with the SI unit $L=1$ m value. Thus, the empirically correlated outputs become independent of the arbitrary scaling factor $L=\alpha$. Therefore, it seems possible to argue that the seemingly arbitrary choice of the value of $L=1$ m is a choice WLOG as for the empirically correlated output.

However, as for the time of writing of this chapter, this concept or argument is not very well understood and quite possibly may turn out to be an erroneous argument. It is critical for the self-consistency of the proposed model (2.3) for it to be independent of the magnitude choice of the periodic lengthscale $L$. The desired output of the proposed conformally flat model (see Section 2.4) within an order of magnitude as compared with the Planck data [13] in addition to the Tully-Fisher relation [14] and other desired simultaneous implications as tabled makes it; as far as the author estimates more likely than not that the proposed conformally flat dS model is not a result of numerical conspiracy.

3.2 Covariance

Another source of error of this model is the matter of covariance. It was, however, shown in [10] that the result (Eq. (12)) for the conformally flat dS case only applies to all inertial reference frames. It is as of today unclear whether or not this relation and its immediate implications thereafter also hold true for all non-inertial reference frames. The inherent proposed physical property of virtual quantization by exploiting the loop periodicity by definition, as well as the proposed physical principle of said vacuum diagram loop divergence treatment by analytic continuation is proposed to be true in general, that is generally covariant. As far as the author of this chapter understands, it is not strictly necessary for the proposed model and principles to be generally covariant, that also holds true for all non-inertial reference frames since all the relevant data [13] were taken with the solar system as a (very good approximation) inertial frame of reference.

3.3 Possible methods of falsification

The following section will give a list of possible methods of falsification of this proposed physical principle. For instance, rigorously showing $\mathrm{one}$ of the following would be a valid falsification as far as the author of this chapter understands:

• Rigorously show that the default magnitude choice of the $L$ periodic lengthscale does not maintain generality.

• Rigorously show that the proposed physical principle of vacuum diagram loop divergence treatment by analytic continuation will in a vacuum diagram case yield a pole. For example by getting a harmonic series ($s=1$) in the analysis, which does not allow for analytic continuation.

• Rigorously show that the proposed inherent virtual quantization is not mathematically or physically consistent.

• Rigorously show that an as of today unknown implication or implications of said proposed principle gives empirical predictions that turn out to be incompatible with present or future data with statistical significance.

• Rigorously show that any one of the known implications is incompatible with present or future data with statistical significance. Given that the exact value of the geometrical proportionality constant is as of today yet to be determined, more than one order of magnitude deviation between these implications and any data with statistical significance will be considered critical.

As outlined in Section 3.2, it is not thought that the general showing of a lack of covariance of the proposed model would be a valid method of falsification since the conformally flat dS case (Eq. (12)) was only claimed [10] to hold for all inertial reference frames, and it remains unclear whether or not it is generally covariant.

3.4 Concrete predictions

The proposed model yields the following concrete predictions:

• Every vacuum diagram virtual loop divergence will $\mathit{\text{without}}\mathit{\text{exception}}$ be treatable by analytic continuation, and the result will be identical to the physical result and hold true at least for all inertial reference frames.

• The quantization of virtual particle loops is an inherent physical property, and thus must exists independently of the presence of a Casimir cavity.

In addition to this, there are the following concrete predictions from its immediate implications:

• According to this proposed model, the black hole physical singularity is indeed unphysical, consequently it is plausible that black hole interiors consist of other cosmologies, and thus that our own cosmology may be identical to a black hole’s interior.

• All the cosmological information is holographically embedded at its surface, and thus the closed region considered must expand with time at all times. Black holes $\mathit{do}\mathit{\text{not}}$ shrink.

4. Conclusions

This chapter proposes the principle of vacuum diagram loop divergence treatment by analytic continuation (what may be referred to as the principle of analyticity) as a method for developing a tentative intuition on how to treat these vacuum diagram loop divergences in general. It is proposed that the virtual quantization is an inherent physical property due to periodic boundary conditions, and thus may exist independently of the presence of a Casimir cavity and can only be manipulated in magnitude by it. It is shown that by using this proposed physical principle, that it is possible to associate a physical value to the desired vacuum diagram Feynman loop integral, which would be identical to its physical value. The said proposed physical principle is referenced [10] to its application in the conformally flat dS case, which seems as far as the author of this chapter understands to be internally self-consistent for all inertial reference frames. It remains unclear whether or not it is covariant in general. Some immediate and simultaneous implications are restated for the sake of clarity and completeness. Some methods of error and falsification are then discussed, and concrete predictions stated.

Conflict of interest

The author declares no conflict of interest.

A.

Here some quick derivations in Ref. to [10] are shown. The law of energy conservation of the system considered is

$$\mathit{dQ}=\mathit{TdS}+\mathit{pdV}=0$$E13

Note that the internal energy $\mathit{dU}=\mathit{TdS}$ does not need to originate from the black hole mass due to the extra term, that is. $\mathit{TdS}\ne \mathit{dM}$; hence, it is possible to avoid evaporation. Where the Hawking temperature $T$ and the Bekenstein entropy $S$ are given by [17].

$$T=\frac{\mathrm{\hslash }{c}^3}{8\mathit{\text{πkGM}}}$$E14

$$S=\frac{\mathit{k\pi }{R}^2}{l_p^2}$$E15

Where $l_p$ is the Planck length. Using the negative pressure up to a geometrical proportionality constant expected to be of order $O\left(-1\right)$

$$p_{\mathit{dS}}\propto -\frac{{\mathrm{\hslash }}^2{c}^6}{G}$$E16

And solving (Eq. (13)) for a sphere of radius $R$ yields two global nonzero minima

$$M^2\propto {\left(\frac{c}{\mathrm{\hslash }G}\right)}^2$$E17

Presumably corresponding to the matter vs. antimatter solutions.

Finally, the equivalence principle combined with the Friedmann equations [18] imply

$$g=a_{\mathit{dS}}=\sqrt{\frac{{\mathrm{\Lambda }}_{\mathit{dS}}}{3}}{c}^2\propto \mathrm{\hslash }{c}^3$$E18

which may be solved for a rotating massive body to give

$$v^4\propto \mathrm{\hslash }{\mathit{Gc}}^{3}M$$E19

Note that it is possible to re-derive (Eq. (17)) by setting $v=c$ up to a geometrical proportionality constant expected to be of $O\left(-1\right)$$\text{independently}$ of the usage of holography (Eq. (15)).

Availability of data and materials

No new data was generated during this study.

Abbreviations

dS	de-Sitter
vs.	versus
i.e.	that is
QFT	Quantum field theory
SI	System international
WLOG	Without loss of generality
&	and

Nomenclature

EOS	Equation of state
EOM	Equation of motion
default	magnitude without external manipulation