Bi-local fields interacting with a constant electric field and related problems including the Schwinger effect

The bi-local fields are known to be an original form of non-local fields extended in spacetime proposed by H. YukawaYukawa-1 -Yukawa-3 . In the beginning, this theory had been expected as a drastic approach to save the problems in local field theories such as the problems of divergence and those of unified description for many elementary particles, etc. According to the development of this theory, however, the bi-local fields established the standpoint as an effectiver approach to the relativistic two-body bound systemsTakabayasi such as mesons in QCD. The resultant formulation of bi-local systems, the classical counterparts of the bi-local fields, had a similar structure to the string modelsBL-Review .

The bi-local systems have a structure of constrained dynamical systems based on the reparameterization invariance of the time parameters of constituent particles. The structure of constrained dynamics is simpler than that of the string model; however, because of this reason, it was not easy to formulate the interaction of the bi-local systems with other fields. In the attempts of those interacting bi-local systems, the scattering amplitudesTanaka , the form factor by external fieldsNamiki and so on were studied. Through the study on a three vertex function of the bi-local fields, a prototype of the Witten type of vertex function in the string model was also proposedGoto-Naka . We also point out the recent viewpoint relating the bi-local fields and the collective bi-local fields out of a higher-spin gauge theory in AdS spacetimeAdS-BL .

In this paper, we study the bi-local systems of two particles carrying the charges $g_{(1)}(>0)$ and $g_{(2)}(<0)$ under a constant electric field. We there take notice of that this type of interaction is relevant to the problem of dissociation and pair production of the bi-local systems by the electric field. In the next section, we review the formulation of free bi-local systems bounded by a relativistic Hooke type of potential.

In Sec.3, a formulation of a bi-local system under a constant electric field is given. The gauge structure of the bi-local system is represented by two constraints: the on-mass shell condition corresponding to a master wave equation of this system and a supplementary condition eliminating redundant degrees of freedom. The consistency of those constraints are spoiled by the presence of the electric field in general; and, we try the modification of those constraints so as to recover the consistency.

In Sec.4, the structure of the physical states satisfying the mass shell condition with the supplementary condition are studied. Therein, the dynamical variables of the system are resolved into the components lie in $\|$ and $\perp$ planes, where the $\|$ is the plane spanned by the direction of time ($\bm{e}^{0}$) and that of electric field $E\bm{e}^{1}$, and $\perp$ is the $(\bm{e}^{2},\bm{e}^{3}$) plane orthogonal to $\|$ ¹¹1We write the basis of 4-dimensional spacetime as $\bm{e}^{\mu}\,(\mu=0,1,2,3)$, to which diag$(\eta^{\mu\nu}=\bm{e}^{\mu}\cdot\bm{e}^{\nu})=(-+++)$. . In terms of those components, we show the way to construct a set of consistent constraints under the presence of $E\neq 0$. Under those constraints, in Sec.5, the structure of the physical states are studied in view of that the $\|$ components of modified center of mass momenta in the mass-shell condition appear in a form of Hamiltonian for a repulsive harmonic oscillatorRHO-PRD .

Within the physical states, the ground state $|0_{g}^{\|}\rangle$ for the internal variables in the $\|$ plane is particularly interesting, since it depends on the charges of respective particles $g_{(i)}(i=1,2)$ and on $E$. Furthermore, if the bi-local systems are embedded in the two-dimensional $(\bm{e}^{0},\bm{e}^{1}$) spacetime, the $\|$ plane, the excited states of those systems are constructed only on the ground state $|0_{g}^{\|}\rangle$ .

In Sec.6, we study the transition amplitude between the ground state $|0_{g}^{\|}\rangle$ for $E_{1}=0$ and one for $E_{2}\neq 0$. The analysis on this transition gives us a knowledge on the stability of the system under the $E$.

In Sec.7, the discussion on the Schwinger effect for the bi-local fields is given. The bi-local systems are neutral or charged systems according as $|g_{(1)}|=|g_{(2)}|$ or $|g_{(1)}|\neq|g_{(2)}|$ respectively. The analysis of the Schwinger effect in the case of $|g_{(1)}|=|g_{(2)}|$ will give us the knowledge on the dissociation of bound states by the $E$.

Section 8 is devoted to the summary of our results. In the Appendices, some mathematical problems used in the text are discussed: in particular in Appendix C, the transition amplitudes between the ground states of different $E$’s are evaluated in detail for the analysis in Sec.6.

A commonly used approach to the bi-local system, the two particle system bounded by a confining potential, is to start from the action

where $x_{(i)}(\tau)^{\mu}~{}(i=1,2)$ are the coordinates of respective particles; and, the interaction terms $V_{(i)}(x)~{}(i=1,2)$ are set as functions of $x\equiv x_{(1)}-x_{(2)}$ to ensure the translation invariance of this system. The $e_{(i)}(\tau),(i=1,2)$ are einbeins, which guaranty the invariance of $S$ under the $\tau$ reparametrization. When this invariance is consistent with the dynamics of bi-local system, the interaction via $V_{(i)}$’s can be described as an action at a distance between $x_{(i)}(\tau)\,(i=1,2)$ with the same $\tau$.

Then varying the action with respect to $e_{(i)}$, we obtain the constraints

where $p_{(i)\mu}=\frac{\delta S}{\delta x_{(i)}^{\mu}}=\frac{1}{e_{(i)}}\dot{x}_{(i)\mu},(i=1,2)$ are momenta conjugate to $x_{(i)}^{\mu}$’s; and we have use the symbol $\approx$ for the constrained equations. The Eq.(2) means that $m_{(i)}\equiv c^{-1}\sqrt{V_{(i)}}~{}(i=1,2)$ are effective masses of respective particles under the interaction.

In what follows, we confine the discussion to the bi-local system of equal mass particles by taking the application in a latter section into account; and, we put

where $c$ is the speed of light. The action (1) tends to the sum of actions of free mass $m$ particles in the limit $\kappa\rightarrow 0$. In this equal mass case, one can introduce the center of mass momentum $P=p_{(1)}+p_{(2)}$ and the relative momentum $p=\frac{1}{2}(p_{(1)}-p_{(2)})$ conjugate to $X=\frac{1}{2}\left(x_{(1)}+x_{(2)}\right)$ and $x$, respectively. In terms of those variables, the constraints (2) can be written as

In the classical mechanics, because of $\{H,T\}=\kappa P\cdot x$, the equations (4) and (5) don’t form a set of first class constraints. Even if we add $P\cdot x\approx 0$ as a secondary constraint, the algebras $\{T,P\cdot x\}=-2\kappa P^{2}$ leads to inconsistent secondary constraint $P^{2}\approx 0$.

In the quantized theories, however, this situation is successfully changed by introducing the oscillator variables $(\hat{a}^{\mu},\hat{a}^{\mu{\dagger}})$ defined by

which satisfy the commutation relation $[\hat{a}_{\mu},\hat{a}_{\nu}^{\dagger}]=\eta_{\mu\nu}$. In terms of those oscillator variables, Eq.(4) can be read as the wave equation

where $\alpha^{\prime}=\frac{1}{8\hbar\kappa}$ and $(m_{0}c)^{2}=4(mc)^{2}+16\hbar\kappa$. Then, Eq.(5) should be regarded as a subsidiary condition on $\Phi$ in some way. To avoid the problem of secondary constraints, we treat Eq.(5) as the expectation value $\langle\Phi|\hat{T}|\Phi\rangle=0$ in analogy with the Gupta-Bleuler formalism in Q.E.D., which suggests to put

Then, because of $[\hat{H},\hat{T}^{(+)}]=-\frac{1}{\alpha^{\prime}}\hat{T}^{(+)}$, the subsidiary condition (8) is compatible with the on-mass shell condition (7).

The states generated by $\{\hat{a}^{\mu{\dagger}}\}$ with the ground state $\hat{a}^{\mu}|0\rangle=0\,(\langle 0|0\rangle=1)$ is an indefinite metric functional space due to $[\hat{a}^{0},\hat{a}^{0{\dagger}}]=-1$. The subsidiary condition (8) can remove the excitation of $\hat{a}^{0{\dagger}}$, since Eq.(8) implies $\hat{a}^{0}|\Phi\rangle=0$ in the rest frame of the center of mass momentum $P=(P^{0},\bm{0})$. This also means that the particles characterized by Eqs. (7) and (8) have $(\mbox{mass})^{2}c^{2}=\frac{1}{\alpha^{\prime}}\sum_{i=1}^{3}n_{i}^{2}+(m_{0}c)^{2}~{}(\,n_{i}=0,1,2,\cdots\,)$ ²²2 We note that the functional space characterized by Eqs.(7) and (8) forms a non-unitary representation of the Lorentz group, which guarantees the manifest covariance of the formalism. In some cases, however, one may treat $\langle\Phi|T|\Phi\rangle=0$ as $T^{(-)}|\Phi\rangle\equiv P\cdot a^{\dagger}|\Phi\rangle=0$. In this case, the operator $a^{0{\dagger}}$ plays a role of annihilation operator; and, the positive definite metric functional space generated by the action of $(b^{\mu{\dagger}})=(a^{0},a^{i{\dagger}})$  $(i=1,2,3)$ on the ground state $b^{\mu{\dagger}}|0_{b}\rangle=0\,(\,\langle 0_{b}|0_{b}\rangle=1\,)$ gives rise to a unitary representation of the Lorentz group in a positive definite metric functional space. The difference of those two representations reflects in the phenomenological properties of the bi-local fields. In this paper, we require Eq.(8) as the subsidiary condition on the bi-local fields to maintain the the manifest covariance of the formalism. It should be noticed that the condition (8) does not exclude non-zero excitation $\{n_{1}\}$ even when the bi-local system exists in two-dimensional $(\bm{e}^{0},\bm{e}^{1})$ spaceteime; this fact is not trivial for the bi-local system interacting with a constant electric field $E\bm{e}^{1}$ as will be discussed in Sec.4.

It is not difficult to extend formally the action (1) to a bi-local system interacting with external electric fields. Let $g_{(i)}\,(i=1,2)$ be the electric charges of the particle with the position variables $x_{(i)}\,(i=1,2)$ in (Fig.1); we set $g_{(1)}\!>0$ and $g_{(2)}\!<0$ for the later analysis. Then, with the gauge potential $A^{\mu}$, the extended action should be

from which the momenta conjugate to $x_{(i)}\,(i=1,2)$ are given as $p_{(i)}=\frac{1}{e}\dot{x}_{(i)}+\frac{g_{(i)}}{c}A(x_{(i)})\,(i=1,2)$. Further, the constraint (2) in this action becomes

$(i=1,2)$, where $\pi_{(i)}=p_{(i)}-\frac{g_{(i)}}{c}A(x_{(i)})\,(i=1,2)$. From now on, we discuss the case in which a constant electric field $E$ is applied to the $\bm{e}^{1}=(0,1,0,0)$ direction; and we put the gauge potentials so that

Then one can write ³³3Throughout this paper, we use the notation of decomposition for any four vector $(f^{\mu})$ such as $f^{\|}=f_{\|}=(f^{0},f^{1}),\,\tilde{f}^{\|}=\tilde{f}_{\|}=(f^{1},f^{0})$, and $f_{\perp}=f^{\perp}=(f^{2},f^{3})$.

Those $\pi_{(i)}^{\prime}s$ can be decomposed into the center of mass and the relative components by some ways such as

where $\tilde{g}_{(i)}=g_{(i)}/g\,(i=1,2)$ and $g=\frac{1}{2}\left(g_{(1)}-g_{(2)}\right)$. In the case of $g_{(1)}=-g_{(2)}=g(>0)$, those definitions give rise to $(\tilde{g}_{(1)},\tilde{g}_{(2)})=(1,-1)$ and $(\Pi_{g},\pi_{g})=(\Pi,\pi)$. We note that the $(\Pi_{g},\pi_{g})$ have the meaning of the center of mass and the relative components of $\pi_{(i)}$’s related to the charge distributions in the bi-local system. As needed, one can also use the relations

where $w_{g}=\frac{1}{2}\left(\sqrt{|\tilde{g}_{(1)}|}+\sqrt{|\tilde{g}_{(2)}|}\right)$, $\Delta_{g}=\sqrt{|\tilde{g}_{(1)}|}-\sqrt{|\tilde{g}_{(2)}|}$, and $\tilde{g}_{*}=\sqrt{|\tilde{g}_{(1)}\tilde{g}_{(2)}|})$, which tend $(w_{g},\Delta_{g},\tilde{g}_{*})\rightarrow(1,0,1)$ according as $|g_{(i)}|\rightarrow g$.

Now, for $E\neq 0$, the primary constraints (4) and (5) are modified so that

Here, the ($H_{E\perp},T_{E\perp}$) are the parts of the primary constraints consisting of $\perp$ components of four vectors. Those are independent of $E$; and so,

Meanwhile, the ($H_{E\|},T_{E\|}$) are $E$ dependent parts of the constraints; that is,

It is obvious that the presence of $E\neq 0$ does not alter the $\perp$ components of the canonical variables, though the $\|$ components of the canonical variables are affected by the electric field. As a result, the Poisson brackets of those variables become, for $\mu,\nu=0,1$,

with $\{\Pi_{\|}^{\mu},x_{\|}^{\nu}\}=0,~{}\{\pi_{\|}^{\mu},x_{\|}^{\nu}\}=-\eta_{\|}^{\mu\nu}$. Here, $\Delta\tilde{g}=\tilde{g}_{(1)}+\tilde{g}_{(2)}$ and $\epsilon^{\mu\nu}=-\epsilon^{\nu\mu}\,(\mu,\nu=0,1;\,\epsilon^{01}=1)$.

Because of those non-vanishing structures of the right-hand sides of equations (23)-(25), it is not easy to construct a set of first class constraints such as the $(H,T^{(+)})$ in the case of $E=0$. Further, the on-mass-shell structure of the bi-local system described by $H_{E}=0$ is hard to understand in terms of $(\Pi_{\|},\pi_{\|})$. To overcome this situation, let us introduce a new set of momenta $(\Pi_{[g]\|},\pi_{g\|})$ with

to which one can verify

in addition to $\{\pi_{g\|}^{\mu},\pi_{g\|}^{\nu}\}=0$ and $\{\pi_{g\|}^{\mu},x_{\|}^{\nu}\}=-w_{g}\eta_{\|}^{\mu\nu}$. The equations (27)-(28) mean that the $\Pi_{[g]\|}$ is independent of the relative variables, though there remains the effect of $E\neq 0$ for $\Delta_{g}\neq 0$

Next, rewriting $(H_{E\|},T_{E\|})$ in terms of $(\Pi_{[g]\|},\pi_{g\|})$, we obtain

Using these expressions to the functions of primary constraints $(H_{E\|},H_{E\perp};T_{E\|},T_{E\perp})$, one can verify the following algebras:

from which follows

where

The Eq.(33) says that the time development of $T_{E}$ by the Hamiltonian $H_{E}$ produces secondarily the constraint $T_{x}\approx 0$, which does not form a set of the first class constraints with $T_{E}\approx 0$. To handle this problem, we deal with these constraints so that we first eliminate $T_{E}$ in $H_{E}$ by replacing $T_{E\|}=T_{E}-T_{E\perp}\rightarrow-T_{E\perp}$, and next require an additional constraint $\Pi_{[g]\|}\cdot\tilde{x}_{\|}\approx 0$. In other words, we use $H_{E\|}$ in the modified form

; then, a little calculation leads to

where

Using this expression of $\mathcal{H}_{E}$, the time development of $\Pi_{[g]\|}\cdot\tilde{x}_{\|}$ generates the constraints $\big{(}{\Pi}_{[g]\|}\cdot{x}_{\|},\,{\Pi}_{[g]\|}\cdot{\pi}_{\|},\,{\Pi}_{[g]\|}\cdot\tilde{\pi}_{\|}\big{)}\approx 0$ secondarily according to the algebras

It should be noticed that $T_{x\|}=\frac{w_{g}}{\tilde{g}_{*}}{\Pi}_{[g]\|}\cdot{x}_{\|}\approx 0$ arises as a crew of those secondary constraints; and so, we need to require $T_{x\perp}=P_{\perp}\cdot x_{\perp}\approx 0$ separately.

The equations (38)-(41) also mean that a linear combination of ${\Pi}_{[g]\|}\cdot\tilde{x}_{\|}\sim{\Pi}_{[g]\|}\cdot{x}_{\|}$ is able to form a set of first class constraint with $\mathcal{H}_{E}$. That is, under an appropriate choice of $a_{i}\,(i=1\sim 4)$, the combination

will satisfy the closed algebra $\{\Lambda_{\|},\mathcal{H}_{E}\}=\bar{k}\Lambda_{\|}$. Then the $\Lambda_{\|}\approx 0$ will play a role of the physical state condition which should be compared with $\hat{T}^{(\pm)}|\Phi\rangle=0$ in the free bi-local fields; we must discuss these points in detail.

In the quantized theories, the Poisson brackets must be replaced by the commutation relations according to $[\hat{f},\hat{g}]=i\hbar\{f,g\}$. Then, according to Eqs.(38)-(41), the algebra $[\hat{\Lambda}_{\|},\hat{\mathcal{H}}_{E}]=k\hat{\Lambda}_{\|}$ gives rise to the following system of simultaneous equations:

To write down the solutions for $(a_{i})$, it is convenient to use the notations

in terms of which one can write

Then the solutions can be represented as (Appendix B)

where $N$ is a normalization constant. The $K_{g}$ is the following function of $E$, which has a meaning of an effective coupling constant for $\|$ components of the bi-local systems, with the step limits:

Substituting $(\bm{u}^{(\pm)},\bm{v}^{(\pm)}_{\epsilon})$ for Eq.(48), we obtain

where the double sign $\pm$ corresponds to that of $\bm{n}_{\pm}$. For reference, the $k$ associated with $\hat{\Lambda}^{(\pm)}_{\epsilon\|}$ is given in Eq.(147).

The next step is to introduce the ladder operators $(\hat{a}^{\mu}_{g\|},\hat{a}^{{\dagger}\nu}_{g\|}),\,(\mu=0,1)$ which play the role of the oscillator variables for the $\mathcal{H}_{E\|}$ and the $\hat{\Lambda}^{(\pm)}_{\epsilon\|}$. Remembering $[\hat{\pi}_{g\|}^{\mu},x_{\|}^{\nu}]=-i\hbar w_{g}\eta^{\mu\nu}_{\|}$ and $[\hat{\pi}_{g\|}^{\mu},\hat{\pi}_{g\|}^{\nu}]=0$, this can be done straightforward; and, we obtain

Then one can verify $[\hat{a}^{\mu}_{g\|},\hat{a}^{{\dagger}^{\nu}}_{g\|}]=\eta_{\|}^{\mu\nu}=\eta^{\mu\nu},\,(\mu,\nu=0,1)$ in addition to the step limits

In terms of those ladder operators, with the setting $N=\sqrt{\frac{K_{g}w_{g}}{2\hbar}}$, we obtain the expressions