The cosmic string as a channel for the massive particle teleportation

TL;DR

This paper introduces a novel mechanism for quantum teleportation of massive particles via stable string cusps, expanding understanding of quantum information transfer in theoretical physics.

Contribution

It demonstrates the existence of stable string cusps that enable quantum particle emission and proposes a new teleportation mechanism based on these singularities.

Findings

Existence of stable string cusps during finite time.

Cusps enable emission of captured massive quantum particles.

Proposes a new quantum teleportation mechanism at large distances.

Abstract

Here we prove the existence of a new type of the world-sheet string singularities - the cusps that are stable during the finite time. These singularities make the emission of the captured massive quantum particle possible in the frames of the author's model suggested earlier. In aggregate, we have a new mechanism of quantum teleportation of such particles at large distances.

Equations

$$⟨\mathbf{p}\mid {\hat{\mathbf{V}}}_{\mathbf{a}}(\mathbf{t})\mid {\mathbf{p}}^{\prime }⟩={ϵ}_{\mathbf{a}}{\chi }_{\mathbf{a}}(\mathbf{p}){\chi }_{\mathbf{a}}({\mathbf{p}}^{\prime })\underset{-\infty }{\overset{\infty }{\int }}{\mathbf{e}}^{-\mathbf{i}(\mathbf{p}-{\mathbf{p}}^{\prime })\mathbf{x}(\mathbf{t},\mathbf{s})}\mathbf{g}(\mathbf{s})\mathbf{ds},$$

$${\mathbf{e}}_{\pm }({\xi }_{\pm })=\pm \frac{1}{\varkappa }{\partial }_{\pm }\mathbf{X}({\xi }_{\pm }),$$

$${\hat{\mathbf{E}}}_{\pm }={{\mathbf{e}}_0}_{\pm }{1}_2-\sum _{i=1}^3{{\mathbf{e}}_{\mathbf{i}}}_{\pm }{\mathbf{\sigma }}_{\mathbf{i}}.$$

$${\hat{\mathbf{E}}}_{+}=K{\hat{\mathbf{E}}}_{-}{K}^{+}.$$

$${\partial }_{+}\left(K^{-1}{\partial }_{-}K\right)=0.$$

$$K=\left(\begin{array}{cc}1& 0\\ -{\alpha }_{+}& 1\end{array}\right)\left(\begin{array}{cc}\exp (-\phi /2)& 0\\ 0& \exp (\phi /2)\end{array}\right)\left(\begin{array}{cc}1& {\alpha }_{-}\\ 0& 1\end{array}\right).$$

$${\partial }_{+}{\partial }_{-}\phi$$

$${\partial }_{\pm }{\rho }_{\mp }$$

$${\partial }_{\pm }{\alpha }_{\mp }$$

$$\mathbf{I}$$

$${\mathbf{II}}_1+\mathrm{i}{\mathbf{II}}_2$$

$$\Bigr{(}\,-{\rm Re}\left(\,t_{i1{\pm}}\overline{t}_{i2{\pm}}\right)\,,\quad-{\rm Im}\left(\,t_{i1{\pm}}\overline{t}_{i2{\pm}}\right)\,,\quad\frac{1}{2}\,(-|t_{i1{\pm}}|^{2}+|t_{i2{\pm}}|^{2}\,)\,\Bigl{)}\,,$$

$$T_{\pm }^{\prime }(\xi )+Q_{\pm }(\xi )T_{\pm }(\xi )=0,$$

$$Q_{-}(t,s)=K^{-1}{\partial }_{-}K,Q_{+}(t,s)=-({\partial }_{+}K)K^{-1}.$$

$$T_{\pm }(\xi )⟶{\tilde{T}}_{\pm }(\xi )=T_{\pm }(\xi )B,B\in SL(2,C),B_{ij}=const,$$

$$(\phi ,{\rho }_{\pm },{\alpha }_{\pm })⟶(\tilde{\phi },{\tilde{\rho }}_{\pm },{\tilde{\alpha }}_{\pm }),$$

$$\tilde{\phi }({\xi }_{+},{\xi }_{-})$$

$${\tilde{\rho }}_{\pm }({\xi }_{\pm })$$

$${\tilde{\alpha }}_{\pm }({\xi }_{+},{\xi }_{-})$$

$$Q_{\pm }(\xi )=-{\rho }_{\pm }(\xi ){\mathbf{\sigma }}_{\pm }+{\overline{\rho }}_{\pm }(\xi ){\mathbf{\sigma }}_{\mp }.$$

$${\rho }_{\pm }(\xi )\equiv 0,\xi \in [a_{\pm },b_{\pm }],-\infty \le a_{\pm }<b_{\pm }\le \infty ,$$

$$\mathbf{X}(t,s)={\mathbf{X}}^0+\varkappa \left({\int }_0^{s+t}{\mathbf{e}}_{+}(\eta )d\eta +{\int }_0^{s-t}{\mathbf{e}}_{-}(\eta )d\eta \right).$$

$$\partial_{+}{\bf X}\partial_{-}{\bf X}=-\frac{\varkappa^{2}}{2}\Big{|}\,t_{11+}(\xi_{+})t_{22-}(\xi_{-})-t_{12+}(\xi_{+})t_{21-}(\xi_{-})\Big{|}^{\,2}\,.$$

$$T_{-}({\xi }_{-})\equiv T_{-}^0=-i{\mathbf{\sigma }}_2{T}_{+}^0\equiv -i{\mathbf{\sigma }}_2{T}_{+}({\xi }_{+})|{\xi }_{\pm }|\le b,$$

$$-b+|s|\le t\le b-|s|,$$

$$X_{\mu }(s,t)=X_{\mu }^0+\varkappa t{e}_{\mu },$$

$$\mathrm{Im}{\rho }_{\pm }(\xi )\equiv 0,\xi \in [a_{\pm },b_{\pm }],-\infty \le a_{\pm }<b_{\pm }\le \infty ,$$

Full text

The cosmic string as a channel for the massive particle teleportation

S.V. Talalov

Department of Applied Mathematics, State University of Toliatti,

14 Belorusskaya str., Toliatti, Samara region, 445020 Russia.

svt_19@mail.ru

Abstract

Here we prove the existence of a new type of the world-sheet string singularities - the cusps that are stable during the finite time. These singularities make the emission of the captured massive quantum particle possible in the frames of the author’s model suggested earlier. In aggregate, we have a new mechanism of quantum teleportation of such particles at large distances.

keywords: cosmic strings; cuspidal points.

PACS codes: 03.65 Ge; 11.27 +d

MSC codes: 51P99; 53B50; 81V19; 85A99

We investigate the infinite $4D$ Nambu - Goto (NG) string (see [1], for example) that might be interpreted as a cosmic string in wire approximation [2]. The cuspidal points arising during the string evolution [3, 4] lead to some interesting physical effects: for example, the gravitational wave bursts [5] and the radio bursts [6] in the Universe. One of such effects could be associated with the capture of the massive quantum particle and its subsequent transfer along the string. The possibility of this was demonstrated by the author in the simple non-relativistic model recently [7], where the infinite $4D$ NG string $X_{\mu}(t,s)$ was considered in the gauge $X_{0}\propto t$. The ”spatial part” of the 4-vector $X_{\mu}(t,s)$ – the curve ${\bf x}(t,s)\in E_{3}$ was considered as a source of short-range potential forces acting on the massive non-relativistic quantum particle, for every time $t$. The time-dependent potential is defined by the matrix elements:

[TABLE]

where vector $\bf p$ is momemtum of a particle, the coupling constant ${\epsilon_{a}}<0$ and the Heaviside function $\chi_{a}(\bf p)=\theta(1/a-|\,\bf p|)$. The constant $a\sim 0$ restricts the domain of the considered forces to a small neighborhood of the curve ${\bf x}(s,t)$. The ”form-factor” $g(s)$ is the arbitrary function from the Swarz space ${\mathcal{S}}$ that satisfies the conditions $g(s)\equiv 1$ $\forall\,s\in[-R,R\,]$ (for some $R>>1$) and $0\leq g(s)\leq 1$. The relevance of this ”separable” approximation was justified in [7]. The following effect was demonstrated: the particle, captured at any point ”A” of the string, can be ”transferred” to the other point ”B” of the curve ${\bf x}(t,s)$ if the point ”B” becomes the cuspidal point due to the string evolution. The standard situation when the cusps on the $4D$ string world-sheet form isolated points only (see [8] for example). Thus, we have the collapse of a wave function of the captured particle caused by the appearance of the cusp. Because such point disappears instantly, we have the effect of the mass transfer along the string only: there are no any physical reasons for the emission of the captured particle out from a neighbourhood of the string.

In this article we demonstrate that the cusps on an NG string can exist during the finite time. To prove it we use the author’s approach for the geometrical description of string that has been developed in a number works (see [9, 10, 11] fo details). Let us describe the string dynamics (both regular and cuspy configurations)in Minkowski space-time $E_{1,3}$ in terms of suggested approach briefly. The standard procedure (see [1] for example) leads to the dynamical equations $\partial_{+}\partial_{-}X_{\mu}=0\,$ and constraints $(\partial_{\pm}X_{\mu})^{2}=0$, where the derivatives $\partial_{\pm}=\partial/\partial\xi_{\pm}$ and cone parameters $\xi_{\pm}=s\pm t\,$. So, the objects of our consideration will be time-like world-sheets with orthonormal parametrization. We consider the infinite strings, that’s why we must impose asymptotic conditions on a curve $X_{\mu}=X_{\mu}(s)$ for any value of evolution parameter $t$. The demanded conditions will be formulated later; in the first place we’ll point out to the local structure of the world-sheet. Let us define the pair of light-like and scale-invariant vectors in space $E_{1,3}$:

[TABLE]

where the value $\varkappa$ is an arbitrary (and scale transformed) positive constant. In our opinion, the separation of scale-transformed mode is quite natural here because the scale invariance of the NG theory. It is clear that we can construct the pair of orthonormal bases ${\bf e}_{\nu\pm}(\xi_{\pm})$ in space $E_{1,3}$ that are connected with the introduced vectors ${\bf e}_{\pm}$ by the equalities ${\bf e}_{\pm}=\left({\bf e_{0\pm}}\mp{\bf e_{3\pm}}\right)/2$. Obviously, the definition of these bases has three - parameter arbitrariness in each point $(t,s)$. We will keep this fact in mind and eliminate this ambiguity in the correspondent place. The bases ${\bf e}_{\nu\pm}(\xi_{\pm})$ allow us to define the vector-matrices ${\hat{\bf E}_{\pm}}$:

[TABLE]

The basis ${\bf e}_{-}$ can be transformed into the basis ${\bf e}_{+}$ by the Lorentz transformation; that is why we can define the $SL(2,C)$ - valued field $K(t,s)$ by means of the formula:

[TABLE]

The field $K(t,s)$ is important object for our approach. In accordance with the definition of the vector-matrices ${\hat{\bf E}_{\pm}}$, these matrices satisfy the equalities $\partial_{\pm}{\hat{\bf E}_{\mp}}=0$. Consequenly, the matrix field $K(t,s)$ satisfies to (special) WZWN - equation

[TABLE]

Let us define the comlex-valued functions $\varphi(t,s)$ and $\alpha_{\pm}(t,s)$ by means of Gauss decomposition for the matrix $K(t,s)\in SL(2,C)$:

[TABLE]

In general, these functions are singular because the decomposition (6) is not defined for the points where the principal minor $K_{11}=0$. Let us introduce the regular functions $\rho_{\pm}=(\partial_{\pm}\alpha_{\mp})\,{\exp}(-\varphi)$. The consequence of the equality (5) will be the following PDE

• system:

[TABLE]

From the geometrical viewpoint, the introduction of the function $\varphi$ and the functions $\rho_{\pm}$ is justified by the following formulae for the first $({\bf I})$ and the pair of second (${\bf II}_{i}$, $i=1,2$) fundamental forms of the world-sheet:

[TABLE]

In the formula (9) the function $\beta\in[0,2\pi)$ is an arbitrary parameter and the function $\chi=({\rm Im}\varphi+\arg\rho_{+}+\arg\rho_{-})/2$. These objects will not discussed here. Of couse, the equations (7a) and (7b) could be deduced from the Gauss and Peterson-Kodazzi equations. The cuspidal points correspond to the singularities of the function $\varphi$. These singularities arise when the light-like vectors ${\bf e}_{\pm}$ coinside. Therefore, the vector $\dot{\bf X}=\varkappa\,({\bf e}_{+}+{\bf e}_{-})$ be light-like vector when the condition $\exp(-{\varphi}/{2})=0$ is fullfiled.

The reconstructing formulae for the space components of the vectors ${\bf e}_{\pm}\equiv\pm(1/\varkappa\,)\,\partial_{\pm}{\bf X}(\xi_{\pm})$ are:

[TABLE]

where the index $i$ is correlated with the sign $\pm$ in accordance witn the rule $i=\frac{3\mp 1}{2}$ and the functions $t_{ij\pm}$ are the matrix elements of certain matrices $T_{\pm}(\xi)\in SL(2,C)$. These matrices solve the linear systems

[TABLE]

where

[TABLE]

Let us note that transformation

[TABLE]

corresponds to the Lorentz transformation of space $E_{1,3}$.

Thus, we can reconstruct the tangent vectors ${\bf e}_{\pm}$ through the solutions of the system (7). To do it we must calculate the coefficients $Q_{ij\pm}$ of the systems (11) through the functions $\varphi$, $\rho_{\pm}$, $\alpha_{\pm}$ and add a certain finite number of constants that fix the solutions $t_{ij\pm}$. The singular functions $\varphi$ and $\alpha_{\pm}$ correspond to cuspy string configurations.

The system (7) has a wide group ${\sf G}$ of invariance. Indeed, let the functions $\varphi(\xi_{+},\xi_{-})$, $\rho_{\pm}(\xi_{\pm})$ and $\alpha_{\pm}(\xi_{+},\xi_{-})$ be solutions for the system (7). Then the transformation

[TABLE]

gives the new solution for the system (7) if

[TABLE]

for arbitrary complex-valued functions $f_{\pm}(\xi)$, $g_{\pm}(\xi)$ and such real functions $A_{\pm}(\xi)$ where the conditions $A_{-}^{\prime}A_{+}^{\prime}\not=0$ are fulfilled. Let the subgroup ${\sf G}_{0}\subset{\sf G}$ is defined by the conditions $A_{\pm}(\xi)\equiv\xi$. Performing the factorization procedure for the group ${\sf G}_{0}$, we can to eliminate the uncertainty that arose early when the bases ${\bf e}_{\nu\pm}(\xi_{\pm})$ were defined. Each coset of the group ${\sf G}_{0}$ contains the matrices $Q_{\pm}$ which have the form

[TABLE]

Cosequently, the corresponding matrices $T_{\pm}(\xi)\in SU(2).$ For our subsequent considerations we will take into account these representatives only. We suppose the functions $\rho_{\pm}\in{\mathcal{S}}$; therefore, the world-sheet asymptotically is a planar surface because (9). After factorization procedure 111which is non-covariant gauge fixing we have for the ”time” component of the 4-vector ${\bf X}(t,s)$: $X_{0}(t,s)\equiv X_{0}^{0}+\varkappa\,t$.

Finally, we have the description of the string dynamics in terms: 1) the ”internal” variables $\rho_{\pm}(s\pm t)$ which are invariant under rotations of the space $E_{3}$, space and time translations and scale transformations; 2) the scale transformed constant $\varkappa$; 3) the certain finite number of constants ($X^{0}_{\mu}$, …) that define the immersion of the variables $\rho_{\pm}$ in space and time $E_{3}\times R_{1}$. In addition to a clearer description of the geometry, the suggested approach leads to additional possibilities for the hamiltonization of string dynamics. This question is beyond the topic of this article. Relevant studies, as well as the proofs and the details, can be found in the works [9, 10, 11].

We emphasize that the functions $\rho_{\pm}(\xi)$ are dynamical variables in our approach. They can be arbitrary functions from the space ${\mathcal{S}}$. The restrictions such as $|\rho_{\pm}|>0$ lead to additional constraints for dynamics, moreover they make the definition of ”small variations” $\delta\rho_{\pm}$ ambiguous in the Swarz space. There are no similar constraints here; thus the identities

[TABLE]

should take place for the certain functions $\rho_{\pm}(\xi)$. Note that there are no conformal transformations $\xi_{\pm}\rightarrow\widetilde{\xi}_{\pm}=A_{\pm}(\xi_{\pm})\,$, ($A^{\prime}\not=0$), in this case which could trivialize the coefficients of the forms (9) globally. As a complement, we note also that the possibility to have non-isolated zeros 222which are ignored usually for the functions $\rho_{\pm}(\xi)$ is valid in the pseudo-Euclidean target space only. It is true because the metrics on the (time - like) world-sheet is pseudo-Euclidean. In the Euclidean space we can choose the comlex coordinates $\xi$ and $\overline{\xi}$ to parametrize the ”world-sheet”, instead of the real cone parameters $\xi_{\pm}$ for the pseudo-Euclidean case. As a consequence the complex-analytical function $\rho_{+}(\xi)$ and the complex-(anti)analytical function $\rho_{-}(\overline{\xi})$ can have isolated zeroes only. This fact has an obvious illustration: the gaussian curvature for any soap film in the space $E_{3}$ is either zero everywhere or non-zero everywhere.

As a final step of the procedure which outlined above, let us write out the formula for the explicit reconstruction of the world-sheet ${\bf X}(t,s)$:

[TABLE]

The space components of the light-like vectors ${\bf e}_{\pm}$ are defined by the formulae (10) and the time components are equal to $1/2$ for our gauge. The singularities of the world-sheet – cusps – correspond to the zeroes of the fundamental form ${\bf I}$: in terms of the functions $t_{ij\pm}$ we have the following equality

[TABLE]

Thus the dynamics of a string cusps is defined by the singular solutions of the system (7). So, the special case $\rho_{\pm}\equiv const$ of the equation (7a) is the Liouville equation. The real solutions (that correspond to the string in $3D$ space-time) with singularities of the Liouville equation has been investigated firstly in the work [12]. The system (7) has been investigated firstly in the work [13] as a model of interacting scalar and spinor field in two-dimensional space - time. Corresponding singular solutions have been investigated in the work [14].

Using the formula (16), we are going to prove that a string configuration providing to the conditions (15), can lead to the cusp that will be stable during the finite time. Indeed, the identities $T_{\pm}(\xi)\equiv T_{\pm}^{0}=const$  $\forall\,\xi\in[a_{\pm},b_{\pm}]$ are true for the case (15) in general. The corresponding domain of the world-sheet is a part of a time-like plane. To simplify the consideration we suppose that $a_{\pm}=-b$, $b_{\pm}=b$ in the formulae (15) for the certain constant $b>0$. What happens if the equality

[TABLE]

is valid? In this case the identities $e_{\mu-}(\xi_{-})\equiv e_{\mu+}(\xi_{+})\equiv e_{\mu}$ are true for the certain constant light-like vector $e_{\mu}$. The r.h.s. of the equality (17) is equal to zero identically for all $s,t$ such that $|s\pm t|\leq b$. Thus, if the time ${}^{\prime\prime}t^{\prime\prime}$ satisfies the unequality

[TABLE]

for the certain values of the parameter $s$, the world-sheet degenerates into the light-like line segment

[TABLE]

where the vector $X_{\mu}^{0}$ is a constant vector. Thus, the cusp on the string $X_{\mu}(\cdot,s)$ that arises at the moment $t_{0}=-b$ in the point $X_{\mu}(-b,0)$, will be stable during the period $\Delta t=t_{1}-t_{0}$, before the moment $t_{1}=b$. Note that the s-parametrization the curve $X_{\mu}(\cdot,s)$ for the moments $t_{0}<t<t_{1}$ is degenerated: we have the identity $\partial X_{\mu}/\partial s\equiv 0$ for all $s$ that satisfies inequality $-b+|t|<s<b-|t|$. This fact means that the parameter $s$ may not be the length of the arc of the curve $X_{\mu}(\cdot,s)$ in a singular case in general.

We can ease the condition (15): let the the identity

[TABLE]

is fulfilled only. In accordance with the formulae (10), (11) and (14) the considered string becomes planar on the domain $\xi_{\pm}\in[a_{\pm},b_{\pm}]$. As it is well-known the planar strings can have stable cusps.

What does the stability of the cusp mean from the viewpoint of the model [7] that demonstrates the capture of a massive particle and it transfer into the cuspidal point? In our opinion the cusp that will be stable during the finite time provides not only collapse of the wave function but the subsequent emission of this particle out from string too. Indeed, the cusp moves with the velocity of light $c$, any massive particle moves with the velocity $v<c$. Therefore, if the cusp exists during the finite time, the cuspidal point and the point of the location of the considered particle must diverge with probability $P$ where $0<P<1$. Of course, the corresponding quantity theory for emission effect must be relativistic. We hope to make a more definite conclusions about the value $P=P(a,\Delta t,\dots)$ in the subsequent works.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Acknowledgements. I would like to thank A.K. Pogrebkov who drew my attention to the certain ”pathological” singularities [15] in the field model [13] that motivated me for the research presented here .