Localized States in Quantum Field Theory

TL;DR

This paper clarifies the nature of localized states in quantum field theory, showing they are consistent with Lorentz covariance and causality, and refuting misconceptions about superluminal influence.

Contribution

It provides a detailed quantum field theoretic analysis distinguishing basis states from wave packets, resolving causality concerns and clarifying the implications of the Reef-Schlieder theorem.

Findings

Localized states can exist without violating causality.

Wave packets do not enable superluminal information transfer.

The Reef-Schlieder theorem does not preclude localized states.

Abstract

Localized states in relativistic quantum field theories are usually considered as problematic, because of their seemingly strange (non covariant) behavior under Lorentz transformations, and because they can spread faster than light. We point out that a careful quantum field theoretic analysis in which we distinguish between basis position states and wave packet states clarifies the issue of Lorentz covariance. The issue of causality is resolved by observing that superluminal transmission of information cannot be achieved by such wave packets. Within this context it follows that the Reef-Schlieder theorem, which proves that localized states can exhibit influence on each other over space like distances, does not imply that such states cannot exist in quantum field theory.

Equations

$$I[\phi (x)]=\frac{1}{2}\int \text{mbox}{\mathrm{d}}^{4}x({\partial }_{\mu }\phi {\partial }^{\mu }\phi -m^2{\phi }^2).$$

$${\partial }_{\mu }{\partial }^{\mu }\phi +m^2\phi =0.$$

$$H=\int \text{mbox}{\mathrm{d}}^3\mathbf{x}(\Pi \dot{\phi }-\mathcal{L})=\frac{1}{2}\int \text{mbox}{\mathrm{d}}^3\mathbf{x}({\Pi }^2-{\partial }^i\phi {\partial }_i\phi +m^2{\phi }^2),$$

$$\{\phi (t,\mathbf{x}),\Pi (t,{\mathbf{x}}^{\prime })\}={\delta }^3(\mathbf{x}-{\mathbf{x}}^{\prime }),$$

$$\{\phi (t,\mathbf{x}),\phi (t,{\mathbf{x}}^{\prime })\}=0,\{\Pi (t,\mathbf{x}),\Pi (t,{\mathbf{x}}^{\prime })\}=0,$$

$$\dot{\phi }=\{\phi ,H\},\dot{\Pi }=\{\Pi ,H\},$$

$$\phi (x)=\int \frac{\text{mbox}{\mathrm{d}}^{4}k}{(2\pi {)}^4}c(k)\delta (k^2-m^2){\mathrm{e}}^{-ikx}=\int \frac{\text{mbox}{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}\left(\tilde{a}(\mathbf{k}){\mathrm{e}}^{-ikx}+{\tilde{a}}^{\ast }(\mathbf{k}){\mathrm{e}}^{ikx}\right),$$

$$\tilde{a}(\mathbf{k})=\frac{1}{2\pi }c({\omega }_{\mathbf{k}},\mathbf{k}),{\tilde{a}}^{\ast }(\mathbf{k})=\frac{1}{2\pi }c(-{\omega }_{\mathbf{k}},\mathbf{k}).$$

$$[\phi (t,\mathbf{x}),\Pi (t,{\mathbf{x}}^{\prime })]=i{\delta }^3(\mathbf{x}-{\mathbf{x}}^{\prime }),$$

$$[\phi (t,\mathbf{x}),\phi (t,{\mathbf{x}}^{\prime })]=0,[\Pi (t,\mathbf{x}),\Pi (t,{\mathbf{x}}^{\prime })]=0.$$

$$\dot{\phi }=-i[\phi ,H],\dot{\Pi }=-i[\Pi ,H]$$

$$\phi (x)=\int \frac{\text{mbox}{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}\left(\tilde{a}(\mathbf{k}){\mathrm{e}}^{-ikx}+{\tilde{a}}^{†}(\mathbf{k}){\mathrm{e}}^{ikx}\right),$$

$$[\tilde{a}(\mathbf{k}),{\tilde{a}}^{†}({\mathbf{k}}^{\prime })]=(2\pi {)}^{3}2{\omega }_{\mathbf{k}}{\delta }^3(\mathbf{k}-{\mathbf{k}}^{\prime }),$$

$$[\tilde{a}(\mathbf{k}),\tilde{a}({\mathbf{k}}^{\prime })]=0,[{\tilde{a}}^{†}(\mathbf{k}),{\tilde{a}}^{†}({\mathbf{k}}^{\prime })]=0.$$

$$H=\frac{1}{2}\int \frac{\text{mbox}{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}{\omega }_{\mathbf{k}}\left({\tilde{a}}^{†}(\mathbf{k})\tilde{a}(\mathbf{k})+\tilde{a}(\mathbf{k}){\tilde{a}}^{†}(\mathbf{k})\right)$$

$$=\int \frac{\text{mbox}{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}{\omega }_{\mathbf{k}}\left({\tilde{a}}^{†}(\mathbf{k})\tilde{a}(\mathbf{k})+\frac{\delta (0)}{2}\right).$$

$$\tilde{a}(\mathbf{k})|0⟩=0,$$

$$⟨0|H|0⟩=\frac{1}{2}\int \frac{\text{mbox}{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}{\omega }_{\mathbf{k}}\delta (0)=E_0=\infty .$$

$$|\Psi ⟩=\sum _n\int \frac{\text{mbox}{\mathrm{d}}^3{\mathbf{k}}_1\text{mbox}{\mathrm{d}}^3{\mathbf{k}}_2...\text{mbox}{\mathrm{d}}^3{\mathbf{k}}_n}{(2\pi {)}^{3n}2{\omega }_{{\mathbf{k}}_1}...2{\omega }_{{\mathbf{k}}_n}}\tilde{g}(t,{\mathbf{k}}_1,{\mathbf{k}}_2,...,{\mathbf{k}}_n){\tilde{a}}^{†}({\mathbf{k}}_1){\tilde{a}}^{†}({\mathbf{k}}_2)...{\tilde{a}}^{†}({\mathbf{k}}_n)|0⟩,$$

$$|\Psi ⟩=\int \frac{\text{mbox}{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}\tilde{g}(t,\mathbf{k}){\tilde{a}}^{†}(\mathbf{k})|0⟩.$$

$$i\frac{\partial |\Psi ⟩}{\partial t}=H|\Psi ⟩,$$

$$i\frac{\partial \tilde{g}(t,\mathbf{k})}{\partial t}=({\omega }_{\mathbf{k}}+E_0)\tilde{g}(t,\mathbf{k}),$$

$$\tilde{g}(t,\mathbf{k})={\mathrm{e}}^{-i({\omega }_{\mathbf{k}}+E_0)t}\tilde{g}(\mathbf{k}).$$

$$⟨\Psi |\Psi ⟩=\int \frac{\text{mbox}{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}{\tilde{g}}^{\ast }(t,\mathbf{k})\tilde{g}(t,\mathbf{k}),$$

$$i\frac{\partial \tilde{g}(t,\mathbf{k}}{\partial t}={\omega }_{\mathbf{k}}\tilde{g}(t,\mathbf{k}),$$

$$\tilde{g}(t,\mathbf{k})={\mathrm{e}}^{-i{\omega }_{\mathbf{k}}t}\tilde{g}(\mathbf{k}).$$

$$|\tilde{\mathbf{k}}⟩={\tilde{a}}^{†}(\mathbf{k})|0⟩,⟨\tilde{\mathbf{k}}|=⟨0|\tilde{a}(\mathbf{k}).$$

$$⟨\tilde{\mathbf{k}}|\Psi ⟩=⟨0|\tilde{a}(\mathbf{k})\int \frac{\text{mbox}{\mathrm{d}}^3{\mathbf{k}}^{\prime }}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}^{\prime }}\tilde{g}(t,{\mathbf{k}}^{\prime }){\tilde{a}}^{†}({\mathbf{k}}^{\prime })|0⟩=\tilde{g}(t,\mathbf{k}),$$

$$|\tilde{\mathbf{x}}⟩=\phi (0,\mathbf{x})|0⟩={\phi }^{(+)}(0,\mathbf{x})|0⟩\equiv {\tilde{a}}^{†}(\mathbf{x})|0⟩,$$

$$⟨\tilde{\mathbf{x}}|=⟨0|\phi (0,\mathbf{x})=⟨0|{\phi }^{(-)}(0,\mathbf{x})\equiv ⟨0|\tilde{a}(\mathbf{x}).$$

$$⟨\tilde{\mathbf{x}}|\Psi ⟩=⟨0|{\phi }^{(-)}(0,\mathbf{x})|\Psi ⟩=\int \frac{\text{mbox}{\mathrm{d}}^3\mathbf{k}}{(2\pi {)}^{3}2{\omega }_{\mathbf{k}}}{\mathrm{e}}^{-i\mathbf{k}\mathbf{x}}\tilde{g}(t,\mathbf{k})=\tilde{f}(t,\mathbf{x}),$$

Full text

Localized States in Quantum Field Theory

Matej Pavšič

Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia

e-mail: matej.pavsic@ijs.si

Abstract

Localized states in relativistic quantum field theories are usually considered as problematic, because of their seemingly strange (non covariant) behavior under Lorentz transformations, and because they can spread faster than light. We point out that a careful quantum field theoretic analysis in which we distinguish between basis position states and wave packet states clarifies the issue of Lorentz covariance. The issue of causality is resolved by observing that superluminal transmission of information cannot be achieved by such wave packets. Within this context it follows that the Reef-Schlieder theorem, which proves that localized states can exhibit influence on each other over space like distances, does not imply that such states cannot exist in quantum field theory.

Keywords: Relativistic wave packet, Quantum field theory, Localized states, Position operator, Causality

1 Introduction

In non relativistic quantum mechanics and quantum field theories, states can be represented by wave functions in configuration space. In the case of one particle states, wave function is the probability amplitude of finding the particle at a position ${\bm{x}}$. In the literature there has been a debate whether the analogous is possible in relativistic quantum mechanics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and quantum field theory [12, 13, 14, 15]. Initially, when extending quantum mechanics to incorporate relativity, the subject of investigation was the wave function satisfying the Klein-Gordon equation. Three main difficulties were encountered:

(i) The probability density was found to be either positive or negative.

(ii) Position operator [1, 2, 3, 4, 5, 6, 7, 8, 11] contained an extra term, which spoiled Lorentz covariance of such an operator.

(iii) Relativistic wave packets can spread faster than light, which has been interpreted as violation of causality [16, 17, 18, 19, 20, 21, 5, 15, 22].

With the advent of second quantization, the difficulty (i) was resolved within the framework of quantum field theory (QFT), in which instead of a wave function satisfying the Klein-Gordon equation, one has an operator-valued non Hermitian field that creates particles and antiparticles of opposite electric charge.

It is usually believed that in QFT states cannot be localized, so that QFT solves the difficulties (ii) and (iii) as well. But such a claim has to be confronted with the fact that relativistic quantum mechanics for positive frequencies is embedded in quantum field theory [23, 24]. A consequence is that the localized states, either as wave packets picked around a certain spatial region, or exactly confined within it, also occur in quantum field theory. We will show why such states are not problematic at all. As is well known [25], the Fock space states with definite momentum, $a^{\dagger}({\bm{p}})|0\rangle$, $a^{\dagger}({\bm{p}}_{1})a^{\dagger}({\bm{p}}_{2})...|0\rangle$, can be superposed by means of complex valued wave packet profiles $g(t,{\bm{p}})$, $g(t,{\bm{p}}_{1},{\bm{p}}_{2})$,…, into the states with indefinite momentum. We choose to put time dependence on $g$, and leave $a^{\dagger}$ independent of time. Even if we consider the case of a Hermitian scalar field $\varphi(x)=\varphi^{\dagger}(x)$ (i.e., if the classical field that we quantize is real), the wave packet profile is in general complex and satisfies $g^{*}g>0$. Similarly, in the case of harmonic oscillator, we quantize the classical real variable $x(t)$, whilst a generic quantum state is a superposition of the states $a^{\dagger}|0\rangle$, $a^{\dagger}a^{\dagger}|0\rangle,...,(a^{\dagger})^{n}|0\rangle$, the superposition coefficients $c_{n}$ being complex and satisfying $c_{n}^{*}c_{n}>0$. The problem of the first quantization in which the probability density can be negative, does not exist in the second quantized, i.e., quantum field theory. What is negative in QFT, is the zero component, $j^{0}$, of the current density $j^{\mu}$, interpreted as a charge current density, whilst the probability density, which is given in terms of a wave packet profile, is always positive [26, 27]. This is a straightforward consequence of the fact that the Hamilton operator is positive definite with respect to the Fock space states, created by the action of $a^{\dagger}({\bm{p}})$ on the vacuum, which is annihilated by $a({\bm{p}})$. Therefore, only positive frequency wave functions occur in quantum field theory, a fact which is often overlooked in the literature.

In an appropriate normalization [26, 27], a wave packet profile $g(t,{\bm{p}})$ gives the scalar product $\int\mbox{\rm d}^{3}{\bm{p}}\,g^{*}(t,{\bm{p}})g(t,{\bm{p}})$. It can be Fourier transformed into a position space wave packet profile $f(t,{\bm{x}})$, giving $\int\mbox{\rm d}^{3}{\bm{x}}f^{*}(t,{\bm{x}})f(t,{\bm{x}})$. The absolute square $|f(t,{\bm{x}})|^{2}=f^{*}(t,{\bm{x}})f(t,{\bm{x}})$ gives the probability density of finding the particles at the position ${\bm{x}}$. Similarly, the corresponding momentum space creation and annihilation operators, $a^{\dagger}({\bm{p}})$, $a({\bm{p}})$, can be Fourier transformed into the position space operators111 For simplicity we use here the same symbol $a$ for the Fourier transformed operators as well. $a^{\dagger}({\bm{x}})$, $a^{\dagger}({\bm{x}})$, with $a^{\dagger}({\bm{x}})|0\rangle$ being a state with the particle at position ${\bm{x}}$. The wave packet profile at, say $t=0$, is then $f(0,{\bm{x}}^{\prime})=\delta^{3}({\bm{x}}^{\prime}-{\bm{x}})$.

In the absence of interaction it makes sense to consider single particle states only. As any multiparticle state, also a single particle state $\int\mbox{\rm d}^{3}{\bm{x}}\,f(t,{\bm{x}})a^{\dagger}({\bm{x}})|0\rangle$ satisfies the Schrödinger equation with the Hamilton operator of the considered quantum field theory (i.g., that of a scalar field), and it turns out, as already pointed out in Ref. [23, 24], that $f(t,{\bm{x}})$ satisfies the Klein-Gordon equation for positive frequencies. Relativistic quantum mechanics for positive frequencies (energies) is thus automatically embedded in QFT, which therefore inherits all the issues concerning state localization and causality. As pointed out by Valente [28], there are several distinct concepts of causality in the literature, and not all of them imply faster than light transmission of information which only can lead to causality paradoxes. The ‘causality’ used in algebraic (axiomatic) quantum field theory [29] as one of the axioms is of such a kind that its violation is not problematic. Consequently, the Reeh-Schlieder [30] theorem does not violate relativistic causality [28] and hence does not imply that states cannot be localized in a finite region.

The initially $\delta$-function wave packet evolves with time as the relativistic Green function $G(t,x;0,0)$ considered in Refs. [27, 23]. This has to be taken into account when transforming $f(0,{\bm{x}})$ into another Lorentz frame. It comes out that if at the initial time $t=0$ in a Lorentz frame $S$ a particle is localized at ${\bm{x}}=0$, then from the perspective of another Lorentz frame $S^{\prime}$ the same particle is also localized in the same spacetime point. In the case of a boost, the frame $S^{\prime}$ is merely pseudo rotated with respect to $S$, so that both reference frames have the same origin, and thus in $S^{\prime}$ the particle at $t^{\prime}=0$ is localized at ${\bm{x}}^{\prime}=0$. This is a consequence of the properties of $G(t,x;0,0)\equiv G(t,x)$ whose absolute square $|G(t,x)|^{2}$ is singular on the light cone, and zero everywhere else. Initially, the particle is thus localized in the “origin” of the light cone, regardless of the Lorentz reference that it is observed from. At latter time, the particle is localized on the intersection of a hypersurface $\Sigma$ with the light cone. If $\Sigma$ is a simultaneity hypersurface in the frame $S$, it is no longer a simultaneity hypersurface in the frame $S^{\prime}$. Therefore, an observer in $S^{\prime}$ must consider the wave packet on his simultaneity hypersurface $\Sigma^{\prime}$, in order to see how the wave packet is localized in $S^{\prime}$.

If a wave packet is not $\delta$-like, but spread, then its behavior [27, 18, 23] depends on whether its width is smaller or greater than the Compton wavelength. If it is smaller, then its probability density after some time becomes concentrated in the vicinity of the light cone, and not exactly on the light cone as in the case of zero width. If the wave packet width is greater than the Compton wave length, then the wave packet’s probability density is concentrated around the particle’s classical world line. The particle is localized (in the sense of being peaked) on the intersection of a hypersurface with the spacetime distribution of the probability density. The choice of the hypersurface depends on the Lorentz frame in which we observe the wave packet evolution. Nothing unusual happens if we go into another Lorentz frame: the particle wave function is still a wave packet spread around the classical trajectory, which in a different Lorentz frame has a different velocity. It is important to stress that we use the term “localization” in a broad sense, either as (i) point-like localization, (ii) localization in a finite spatial region and vanishing outside, (iii) localization in a finite region decaying outside, and (iv) “effective” localization like a Gaussian wave packet. We will show that none of those kinds of localization is problematic

In this paper we consider wave packet profiles in the free scalar field theory, revise the role of position operator and the behavior of states under Lorentz transformations. We find that localized states are not problematic at all. We also calculate some explicit examples of wave packets for the widths greater and smaller than the Compton wave length. Finally we discuss the issue of causality violation in the cases when the probability density leaks outside the light cone. We argue that in order to violate causality one has to be able to transmit information faster than light, and that this cannot be achieved by means of the wave packets whose centroid position is on, or inside, the light cone. To transmit information, one wave packet is not enough; it is necessary to have a modulated beam of particles, which can be achieved by sending one wave packet after another. Because their centers move with the velocity of light or slower, a train of the wave packets which bears a message, cannot travel faster than light. Some other authors [10, 20] also had similar ideas. However, there could exist other ingenious ways to use relativistic wave packets to send signals faster than light. But since the Compton length is of a subatomic size, such signals could be sent into a very nearby past only, so that no macroscopic effects of the grand father causality paradox could take place.

2 Wave packet profiles in the free scalar field theory

To make the paper self-consistent and to clarify certain confusion regarding state localization, we will review the essential features of the free scalar field theory. Let us consider a real scalar field $\varphi$, $x\equiv(t,{\bm{x}})$, described by the action

[TABLE]

Variation of the latter action with respect to $\varphi(x)$ gives the Klein-Gordon equation

[TABLE]

From the canonically conjugated variables $\varphi(t,{\bm{x}})$, $\Pi(t,{\bm{x}})=\partial{\cal L}/\partial{\dot{\varphi}}={\dot{\varphi}}$, we can construct the Hamiltonian

[TABLE]

where $\partial_{i}\equiv\partial/\partial x^{i}$, $i=1,2,3$.

Using the Poisson bracket relations

[TABLE]

[TABLE]

we find that the equations of motion

[TABLE]

are equivalent to the Klein-Gordon equation (2).

A general solution of the Klein-Gordon equation is

[TABLE]

where $\omega_{\bm{k}}=\sqrt{m^{2}+{\bm{k}}^{2}}>0$, and

[TABLE]

Upon quantization, $\varphi$ and $\Pi$ become operators satisfying

[TABLE]

[TABLE]

The Klein-Gordon equation (2) is now the equation of motion for the operator $\varphi(x)$, and is equivalent to the Heisenberg equations of motion

[TABLE]

that are quantum analog of the classical equations (6).

The quantum field $\varphi(x)$ that solves the Klein-Gordon equation can be expanded according to

[TABLE]

where ${\tilde{a}}({\bm{k}})$ and ${\tilde{a}}^{\dagger}({\bm{k}})$ are operators satisfying

[TABLE]

[TABLE]

The latter commutation relations for ${\tilde{a}}({\bm{k}})$, ${\tilde{a}}^{\dagger}({\bm{k}})$ are consistent with the commutation relations (9),(10) for $\varphi(x)$, $\Pi(x)$.

The Hamilton operator, given by the expression (3), can be rewritten in terms of ${\tilde{a}}({\bm{k}})$, ${\tilde{a}}^{\dagger}({\bm{k}})$:

[TABLE]

If we define vacuum according to

[TABLE]

then the vacuum expectation value of the Hamiltonian is

[TABLE]

A generic state is a superposition of the basis states created by ${\tilde{a}}^{\dagger}({\bm{k}})$:

[TABLE]

where ${\tilde{g}}(t,{\bm{k}}_{1},{\bm{k}}_{2},...,{\bm{k}}_{n})$ is a complex valued wave packet profile.

A single particle state is

[TABLE]

It evolves according to the Schrödinger equation

[TABLE]

where the Hamilton operator is given in Eq. (15). From the latter equation, by using (16) and the commutation relations (13),(14), we obtain the following equation of motion for the wave packet profile [31], p. 162, [18]

[TABLE]

whose solution is

[TABLE]

The scalar product of a single particle state is

[TABLE]

where the zero point energy, $E_{0}$, cancels out. Therefore, from now on we will omit $E_{0}$ in the expression (21), and assume

[TABLE]

[TABLE]

Let us now project a single particle state $|\Psi\rangle$ (Eq. (19)) onto a basis state $|{\tilde{\bm{k}}}\rangle$, defined according to

[TABLE]

We obtain

[TABLE]

where we have taken into account the commutation relation (13) and the vacuum property (16).

We can also project $|\Psi\rangle$ onto a state $|{\tilde{\bm{x}}}\rangle$ defined according to

[TABLE]

[TABLE]

Here $\varphi^{(+)}$ and $\varphi^{(-)}$ are, respectively, the positive and negative frequency part of $\varphi(x)$, given in Eq. (12). We then have

[TABLE]

and

[TABLE]

In Eq. (30) we have the transformations from the amplitude ${\tilde{g}}(t,{\bm{k}})$ to ${\tilde{f}}(t,{\bm{x}})$. The inverse transformation is

[TABLE]

If we insert the latter expression into the scalar product (23), we obtain

[TABLE]

Let us now use Eq. (24), from which we obtain

[TABLE]

Using (30), Eq. (34) gives the well known relativistic Schrödinger equation [19],[27, 26]

[TABLE]

The scalar product is thus

[TABLE]

We can do the calculation in the opposite way and start from Eq. (36). Inserting the expression (30) for ${\tilde{f}}(t,{\bm{x}})$, and using Eq. (24), we have

[TABLE]

Because the right hand side of the latter equation is Lorentz invariant, also the left hand side is Lorentz invariant. This can be also seen if we rewrite the expression (36) in a covariant way as $2i\int\mbox{\rm d}^{3}{\bm{x}}\,{\tilde{f}}^{*}\partial_{0}{\tilde{f}}=2i\int\mbox{\rm d}\Sigma^{0}\,{\tilde{f}}^{*}\partial_{0}{\tilde{f}}=2i\int\mbox{\rm d}\Sigma^{\mu}\,{\tilde{f}}^{*}\partial_{\mu}{\tilde{f}}$, where in this particular Lorentz frame it is $\mbox{\rm d}\Sigma^{\mu}=(\mbox{\rm d}\Sigma^{0},0,0,0)$.

The scalar product so defined is positive, because the wave packet profile ${\tilde{g}}(t,{\bm{k}})$ satisfies the Schrödinger equation (21) with positive energy. This is so because the Hamilton operator (15) is positive definite with respect to the states created by ${\tilde{a}}^{\dagger}({\bm{k}})$, and because the vacuum satisfies ${\tilde{a}}({\bm{k}})|0\rangle=0$. Analogous holds for a Fourier-like transformed wave packet ${\tilde{f}}(t,{\bm{x}})$ and the operators ${\tilde{a}}^{\dagger}({\bm{x}})$, $a({\bm{x}})$, defined in Eqs. (28)–(30).

3 An alternative normalization of the operators and wave packets

If instead of ${\tilde{a}}({\bm{k}})$ and ${\tilde{g}}(t,{\bm{k}})$ we introduce222Such a normalization is also used in the literature, e.g., in the textbook by Peskin [25].

[TABLE]

and analogous for $a^{\dagger}(t,{\bm{k}})$, $g^{*}(t,{\bm{k}})$, then many expressions and derivations become much simpler.

The field operator becomes

[TABLE]

where $a({\bm{k}})$ and $a^{\dagger}({\bm{k}})$ satisfy

[TABLE]

[TABLE]

so that the Hamiltonian is now

[TABLE]

A generic single particle state (19) can be rewritten as

[TABLE]

From the Schrödinger equation (13) it now follows that [18]

[TABLE]

[TABLE]

The Fourier transformed quantities are

[TABLE]

[TABLE]

where $f(t,{\bm{x}})$ satisfies

[TABLE]

Analogous expressions hold for $a^{\dagger}({\bm{x}})$ and $f^{*}(t,x)$.

Using (40),(41) and (46), we find that $a({\bm{x}})$ and $a^{\dagger}({\bm{x}})$ satisfy the following commutation relations:

[TABLE]

[TABLE]

A generic single particle state (43) can be re-expressed as a superposition of the states created by $a^{\dagger}({\bm{x}})$:

[TABLE]

The scalar product becomes

[TABLE]

It is of course Lorentz invariant, though in the above form does not manifestly look so, because $g(t,{\bm{k}})$ and $f(t,{\bm{x}})$ do not have simple Lorentz transformations [27, 26]. But the quantities ${\tilde{g}}(t,{\bm{k}})$ and ${\tilde{f}}(t,{\bm{x}})\equiv{\tilde{f}}(x)$ are scalars:

[TABLE]

where $t^{\prime},{\bm{k}}^{\prime}$ and $x^{\prime}=\Lambda x$ are Lorentz transformed quantities.

The transformation between $g(t,{\bm{k}})$ and ${\tilde{g}}(t,{\bm{k}})$ is simple, namely (38), whilst the transformation between $f(t,{\bm{x}})$ and ${\tilde{f}}(t,{\bm{x}})$ is [27, 26]

[TABLE]

where

[TABLE]

which gives [27, 26]

[TABLE]

The latter transformation can be straightforwardly derived from Eqs. (47), (32) and (38).

The inverse transformations is

[TABLE]

with

[TABLE]

satisfying

[TABLE]

Analogous transformation also holds for the creation/annihilation operators. Denoting ${\tilde{a}}({\bm{x}})\equiv\varphi^{-}(0,{\bm{x}})$, ${\tilde{a}}^{\dagger}({\bm{x}})\equiv\varphi^{+}(0,{\bm{x}})$, we have

[TABLE]

[TABLE]

If in Eq. (52) we express $g(t,{\bm{k}})$ according to (38) and $f(t,{\bm{x}})$ according to (56), we obtain the scalar product in the form (23) or (36), as we should. To recapitulate, the single state scalar product can be expressed in the following four ways:

[TABLE]

where the state is given as

[TABLE]

It contains positive energy basis states only, negative ones are excluded, bacause ${\tilde{a}}({\bm{k}})|0\rangle=0$. The scalar product is positive.

The transformation between ${\tilde{f}}(t,{\bm{x}})$ and $f(t,{\bm{x}})$ is non local. Thus, if ${\tilde{f}}(t,x)$ at $t=0$ is a localized function of ${\bm{x}}$,

[TABLE]

then $f(0,{\bm{x}})$ is a delocalized function of ${\bm{x}}$ according to

[TABLE]

On the contrary, if $f(t,{\bm{x}})$ at $t=0$ is

[TABLE]

then ${\tilde{f}}(0,{\bm{x}})$ is a delocalized function of ${\bm{x}}$:

[TABLE]

Despite being delocalized in ${\bm{x}}$, the latter function is an eigenfunction of the position operator [1, 2, 3, 4, 5, 6, 7, 8], and it represents a state, localized at position ${\bm{x}}_{0}$. But so does the function $f(0,{\bm{x}})$ of Eq. (66), which, as we will see, is also an eigenstate of the position operator. We see that representation of a state in terms of $f(t,{\bm{x}})$ is better adapted for description of a wave packet state, effectively localized within a finite spatial region. In the next section we will discuss properties of the position operator and localized states in free scalar field theory.

4 Position and momentum operator

In previous sections we represented a generic single particle state as a superposition (63) of the basis states, created either by $a^{\dagger}({\bm{x}})$, $a^{\dagger}({\bm{p}})$, or, ${\tilde{a}}^{\dagger}({\bm{x}})\equiv\varphi^{(+)}(0,{\bm{x}})$,${\tilde{a}}^{\dagger}({\bm{p}})$:

[TABLE]

[TABLE]

the corresponding wave packet profiles being

[TABLE]

[TABLE]

Relations among those four kinds of creation operators and wave packet profiles are given in Eqs. (38),(46),(60),(61),(56) and (62).

Let us consider the operator

[TABLE]

which in momentum space reads

[TABLE]

The action of ${\bm{x}}$ on a basis state $|{\bm{x}}\rangle$ gives

[TABLE]

The basis states $|{\bm{x}}\rangle$ are thus eigenstates of the operator ${\hat{\bm{x}}}$, which can therefore be called position operator.

If we act with the operator ${\hat{\bm{x}}}$ on a generic single particle state (63) and make the projection onto $\langle{\bm{x}}|=\langle 0|a({\bm{x}})$ or $\langle{\bm{p}}|=\langle 0|a({\bm{p}})$, we obtain

[TABLE]

[TABLE]

But if we project the same state (63) onto the states $\langle{\tilde{\bm{x}}}|=\langle 0|{\tilde{a}}({\bm{x}})\equiv\langle 0|\varphi^{(-)}(0,{\bm{x}})$ or $\langle{\tilde{\bm{p}}}|=\langle 0|{\tilde{a}}({\bm{p}})$, then we find

[TABLE]

[TABLE]

which are the well known expressions for the action of the Newton-Wigner position operator [1]–[8] on a wave packet profile that satisfies the scalar product given in Eq. (62).

The extra term in Eq. (78) comes from the factor $\sqrt{(2\pi)^{3}\,2\omega_{\bm{p}}}$ in the transformation (38) between $a({\bm{p}})$ and ${\tilde{a}}({\bm{p}})$, or $g(t,{\bm{p}})$ and ${\tilde{g}}(t,{\bm{p}})$. Equation (77) can then be obtained from the relation (30) between ${\tilde{f}}(t,{\bm{x}})$ and ${\tilde{g}}(t,{\bm{p}})$.

Rewritten in terms of ${\tilde{a}}({\bm{p}})=\sqrt{(2\pi)^{3}2\omega_{\bm{p}}}$, the position operator (73) becomes

[TABLE]

where $\partial/\partial{\bm{p}}\equiv\nabla_{\bm{p}}$. Its Fourier transform is then

[TABLE]

We see that the position operator ${\hat{\bm{x}}}$ has a rather cumbersome form if written in terms of ${\tilde{a}}^{\dagger}({\bm{p}})$, ${\tilde{a}}({\bm{p}})$, or ${\tilde{a}}^{\dagger}({\bm{x}})$, ${\tilde{a}}({\bm{x}})$, whilst it has the simple form (72) or (73) if written in terms of $a^{\dagger}({\bm{p}})$, $a({\bm{p}})$, or $a^{\dagger}({\bm{x}})$, $a({\bm{x}})$. Its action on the wave packet profile in the coordinate and the momentum representation, has the simple forms (75) and (76), respectively.

The position operator in the form (72) or (73) is self adjoint with respect to the scalar product (62) expressed in terms of $f(t,{\bm{x}})$ or $g(t,{\bm{p}})$. The same position operator in the form (79) or (80) is self-adjoint with respect to the scalar product (62) expressed in terms of ${\tilde{f}}(t,{\bm{x}})$ or ${\tilde{g}}(t,{\bm{p}})$.

The representation with $a({\bm{x}})$, $a({\bm{p}})$, and its Hermitian conjugates, thus gives simple expressions and it enables the interpretation of $|f(t,{\bm{x}})|^{2}\equiv f^{*}(t,{\bm{x}})f(t,{\bm{x}})$ as the probability density of finding a particle at position ${\bm{x}}$ and time $t$.

Similarly to the position operator, we can define the momentum operator according to

[TABLE]

From the commutation relations (40),(41), we find

[TABLE]

where

[TABLE]

is the particle number operator.

Defining the center of mass position operator,

[TABLE]

we obtain

[TABLE]

where we have used $[{\hat{\bm{x}}}_{T},{\hat{N}}]=0$ and $[{\hat{\bm{x}}}_{T},1]=[{\hat{\bm{x}}},{\hat{N}}^{-1}{\hat{N}}]=0$.

If the position operator ${\hat{\bm{x}}}$ acts on a state with many particles at positions ${\bm{x}}_{n}$ we have,

[TABLE]

But if the center of mass position operator ${\hat{\bm{x}}}_{T}$ acts on the same state, then we have

[TABLE]

[TABLE]

where we have now used the abbreviated notation for the product.

5 Behaviour of states under Lorentz transformations

The states $|{\bm{x}}\rangle$, defined according to (68) are an idealization that cannot be exactly realized in nature. They form the basis states in terms of which a generic single particle state can be expanded:

[TABLE]

If $f({\bm{x}})=\delta^{3}({\bm{x}}-{\bm{x}}_{0})$, then $|\Psi\rangle=a^{\dagger}({\bm{x}}_{0})|0\rangle$, but in general $|\Psi\rangle$ is a superposition (89), or its many particle generalization,

[TABLE]

In this paper we restrict our consideration to the single particle case, though we could as well consider the many particle case.

When considering the behaviour of $|\Psi\rangle$, $f({\bm{x}})$ and $a^{\dagger}({\bm{x}})|0\rangle=|{\bm{x}}\rangle$ under Lorentz transformations we must be careful in determining which kind of transformation we have in mind, passive of active. In the case of a passive transformation, the state $|\Psi\rangle$ remains the same, whilst the components $f({\bm{x}})$ and the basis states $|{\bm{x}}\rangle=a^{\dagger}({\bm{x}})|0\rangle$ change.

In order to see how the expression (89) for a state $|\Psi\rangle$ looks in another Lorentz frame, let us rewrite it in terms of ${\tilde{f}}(t,{\bm{x}})$ and ${\tilde{a}}^{\dagger}(0,{\bm{x}})\equiv\varphi^{\dagger}(0,{\bm{x}})$:

[TABLE]

[TABLE]

[TABLE]

This can be written as

[TABLE]

where $\mbox{\rm d}\Sigma^{\mu}=(\mbox{\rm d}\Sigma^{0},0,0,0)$, and ${\tilde{a}}^{\dagger}({\bm{x}})\equiv{\tilde{a}}^{\dagger}(0,{\bm{x}})\equiv\varphi^{+}(0,{\bm{x}})$.

The quantity ${\tilde{f}}(t,{\bm{x}})\equiv{\tilde{f}}(x)$ transforms under Lorentz transformations as a scalar, ${\tilde{f}}^{\prime}(x^{\prime})={\tilde{f}}(x)={\tilde{f}}(L^{-1}x^{\prime})$, where $x^{\prime}=Lx$, i.e., $x^{\prime\mu}={L^{\mu}}_{\nu}x^{\nu}$. Similarly, also the operator $\varphi^{\dagger}(x)$ transforms as a scalar, $\varphi^{\prime+}(x^{\prime})=\varphi^{+}(x)$. Therefore, expressed in another Lorentz frame, the state (92) reads

[TABLE]

Here $t^{\prime}_{0}$ is the Lorentz transform of the time $t_{0}=0$.

In the case of a boost in the $x^{1}$ direction, we have

[TABLE]

which gives

[TABLE]

Equation (93) then reads

[TABLE]

In the last expression the quantity $\mbox{\rm d}\Sigma^{\prime\mu}=(\mbox{\rm d}\Sigma^{\prime 0},\mbox{\rm d}\Sigma^{\prime 1},0,0)$ represents the same hypersurface element occurring in eq. (92), but expressed in a new Lorentz frame $S^{\prime}$.

Instead of performing in $S^{\prime}$ the integration over the same 3-surface $\mbox{\rm d}\Sigma^{\prime\mu}={L^{\mu}}_{\nu}\mbox{\rm d}\Sigma^{\nu}$ as in the frame $S$, in which $\mbox{\rm d}\Sigma^{\mu}=(\mbox{\rm d}\Sigma^{0},0,0,0)$, we can as well perform the integration over a different 3-surface, whose elements are $\mbox{\rm d}{\bar{\Sigma}}^{\prime\mu}=(\mbox{\rm d}{\bar{\Sigma}}^{\prime 0},0,0,0)$, and not those given in Eq. (95). Then, instead of (93), we have a different state

[TABLE]

[TABLE]

where ${\bar{t}}^{\prime}_{0}=0$. Now ${\bar{t}}^{\prime}_{0}$ is not a Lorentz transform of the time $t_{0}$ at a spatial position ${\bm{x}}$. The expression (97) for $|{\bar{\Psi}}\rangle$ has the same form as the expression (91) for $|\Psi\rangle$. The same steps as in Eq. (91) can also be done in Eq. (97). Writing ${\tilde{a}}^{\prime\dagger}({\bar{t}}^{\prime}_{0},{\bm{x}}^{\prime})\equiv{\tilde{a}}^{\prime\dagger}({\bm{x}}^{\prime})$, we therefore have

[TABLE]

We see that in a new Lorentz frame $S^{\prime}$ we can form a state $|{\bar{\Psi}}\rangle$ in the analogous way as in the old Lorentz frame $S$, by using the transformed wave packet $f^{\prime}(t^{\prime},{\bm{x}}^{\prime})$ and the transformed creation operators $a^{\prime\dagger}({\bm{x}}^{\prime})$. The latter operator creates a particle at the spacetime event $(t^{\prime}=0,{\bm{x}}^{\prime})$, whilst the original operator $a^{\dagger}({\bm{x}})$ creates a particle at $(t=0,{\bm{x}})$, which in general is a different event than $(t^{\prime}=0,{\bm{x}}^{\prime})$, and on different 3-surface.

Let us now investigate how the scalar product transforms under Lorentz transformations:

[TABLE]

Using the Lorentz transformation (94) for $\mbox{\rm d}\Sigma^{0}$ and transforming $\partial_{0}{\tilde{f}}$ according to

[TABLE]

we obtain

[TABLE]

where, according to (94), $\mbox{\rm d}\Sigma^{\prime 1}=-v\mbox{\rm d}\Sigma^{\prime 0}$. Expression (101) is just a particular case of the covariant expression

[TABLE]

if in the reference frame $S$ the hypersurface is $\mbox{\rm d}\Sigma^{\mu}=(\mbox{\rm d}\Sigma^{0},0,0,0)$.

The scalar product is expressed in the frame $S$ according to Eq. (99), and in the frame $S^{\prime}$ according to Eq. (101). In the frame $S^{\prime}$ not only the time like component, but also the spatial component takes place. This is so because in the frame $S^{\prime}$ the hypersurface element $\mbox{\rm d}\Sigma^{\prime\mu}$, over which we integrate, has also space like and not only time like components.

However, in every Lorentz frame we are free to choose a hypersurface over which to perform the integration333 Frame dependent localization has bee considered in Refs [9, 10].. Thus, instead of taking $\mbox{\rm d}\Sigma^{\prime\mu}=(\mbox{\rm d}\Sigma^{\prime 0},-v\mbox{\rm d}\Sigma^{\prime 0},0,0)$, which in $S$ has components $\mbox{\rm d}\Sigma^{0},0,0,0)$, we can take another hypersurface, whose elements in the frame $S^{\prime}$ are $\mbox{\rm d}{\bar{\Sigma}}^{\prime\mu}=(\mbox{\rm d}{\bar{\Sigma}}^{\prime 0},0,0,0)$. The state is then different, namely (97), and the scalar product is then is not that of Eq.  (101), but is

[TABLE]

The latter expression, valid in the frame $S^{\prime}$, has the same form as the expression (99), valid in the frame $S$. Therefore we can proceed as in Secs 2 and 3 and arrive at the scalar product of the form (62), and the relation (56) between $f(t,{\bm{x}})$ and ${\tilde{f}}(t,{\bm{x}})$, in which $\mbox{\rm d}^{3}{\bm{x}}$ is replaced by $\mbox{\rm d}^{3}{\bm{x}}^{\prime}=\mbox{\rm d}{\bar{\Sigma}}^{\prime 0}$, ${\tilde{f}}(t,{\bm{x}})$ by ${\tilde{f}}^{\prime}(t^{\prime},{\bm{x}}^{\prime})$ and $f(t,{\bm{x}})$ by $f^{\prime}(t^{\prime},{\bm{x}}^{\prime})$. Therefore, the scalar product (103) can be written in the form

[TABLE]

where $f^{\prime*}(t^{\prime},{\bm{x}}^{\prime})f^{\prime}(t^{\prime},{\bm{x}}^{\prime})$ is the probability density in the new Lorentz frame.

Despite that the integrals in Eqs. (103) and (104), or in Eq. (99), are equal, the expressions under the integrals, are not equal [26]. For an illustrative discussion see Refs. [32, 33].

6 Wave packet solutions of the relativistic Schrödinger equation

We have seen that a wave packet profile $f(t,{\bm{x}})$ for a single particle state, created by the action of $a^{\dagger}({\bm{x}})$ on the vacuum, satisfies the relativistic Schrödinger equation (48). Recall that we have obtained such equation within the framework of relativistic quantum field theory (QFT). Usually Eq. (48) is considered from the point of view of relativistic quantum mechanics (QM). But the straightforward procedure, displayed in this paper, (see also Refs. [26, 34]) has shown that relativistic QM (restricted to positive norms) is embedded within relativistic QFT, namely, it is associated with single particle wave packet profiles that, as shown in Sec. 2, automatically have positive norms and energies, once a vacuum, satisfying ${\tilde{a}}({\bm{k}})|0\rangle=0$, is chosen.

We will now study wave packet solutions of equation (48). Let initially the wave function be given by

[TABLE]

its Fourier transform being

[TABLE]

The latter state evolves according to Eq. (44), which gives

[TABLE]

A single particle state is thus

[TABLE]

its projection onto a state $\langle{\bm{x}}|=\langle 0|a({\bm{x}})$ being the Green’s function

[TABLE]

For a generic initial wave function $f(0,{\bm{x}})$, different from (105), we have

[TABLE]

As explicitly derived in Ref. [23] (see also [7]), the Green function in one dimension is

[TABLE]

where $K_{1}$ is the modified Bessel function of degree one. Equation (111) is valid for all values of $x$ and $t$.

Al-Hashimi and Wiese [23] also showed that the wave function of a minimal position-velocity uncertainty wave packet can be expressed in terms of the Green function according to

[TABLE]

where $A$ is a normalization constant, and where $\alpha$, $\beta=\beta_{r}+i\beta_{i}$ are constants, related to the wave packet parameters according to

[TABLE]

where $\Delta v^{2}=\langle v\rangle^{2}-\langle v^{2}\rangle$. Taking into account the relation for a minimal position velocity uncertainty wave packet [23],

[TABLE]

we find

[TABLE]

Using Mathematica we have calculated the probability density $|f(t,x)|^{2}$ for various choices of parameters $m$, $\Delta x$ and $\langle v\rangle$. For the parameter $\beta_{i}$, which determines the initial position of the wave packet, we have set $\beta_{i}=0$. In Fig. 1 are shown the plots for $\langle v\rangle=0$, $\langle v\rangle=0,5$, $m=1$, and the wave packet width $\Delta x=1.4$. We see444 We use the extended Planck units [36] (see also Wikipedia [37]) in which $\hbar=c=G=4\pi\epsilon_{0}=1$. that for $\Delta x>\lambda_{c}$, where $\lambda_{c}=\frac{\hbar}{mc}=\frac{1}{m}$ is the Compton wavelength, we have just the usual wave packet solution with the maximum of $|f(t,x)|^{2}$ corresponding to the expectation value of the particle’s classical trajectory (Fig. 1). But if $\Delta x<\frac{1}{m}=1$, then during certain period the wave packet evolves normally, and afterwhile it splits into two wave packets, whose centers move into the opposite directions with the velocity of light. The information about the wave packet expectation velocity is encoded in different intensities of the two branches (Fig. 2).

Inspecting the wave packets of Figs. 1 and 2, it is obvious that when observed from another Lorentz frame nothing unusual happens. In another Lorentz frame they become Lorentz transformed wave packets. If the initial width decreases, then the probability density $|f|^{2}$ becomes higher and higher, as shown in Fig. 3. In the limit of a $\delta$-like localized wave packet at $t=0$, $|f|^{2}$ becomes infinitely high and infinitely narrow, concentrated on the light cone, according to $|f|^{2}=\frac{1}{2}\left(\delta(t-x)+\delta(t+x)\right)$. The event at $t=0$ and $x=0$ at which the particle is initially localized, is, of course, invariant in all Lorentz frames. Thus all observers see the particle localized in the origin of their Lorentz frame. At later times $t>0$ the particle is localized on the intersection of the simultaneity hypersurface with the light cone. For such a limiting state, their is no instantaneous spreading of the probability density of the sort considered in Refs. [19, 18, 16]. We thus see that the relativistic wave packet in the limit of the $\delta$-like initial localization in fact remedies the non relativistic case, in which an infinitely thin wave packet, exactly localized at $t=0$, spreads over all space at arbitrarily small $t>0$.

We have also seen that the relativistic expression (112), derived from (109), describes wave packets of any velocity, including zero velocity. Thus even a particle moving with zero velocity is described by the relativistic wave packet. The non relativistic wave packet is obtained from expression (110) in the approximation $m^{2}\gg{\bm{p}}^{2}$ in which we neglect higher momenta. Equivalently, it is obtained from expression (112) if the wave packet width $\Delta x$ is large in comparison with the Compton length.

The case in which at $t=0$ a wave is not a minimal position-velocity wave packet, but an exactly localized (rectangular) wave packet, was considered by Karpov et al. [34]. It was found that such wave packet is a superposition of two non local wave packets moving in the opposite directions with the velocity of light. Initially this gives a rectangular localized wave packet, which immediately delocalizes at $t>0$. This is similar to the behavior of a minimal position-velocity wave packet, whose width $\Delta x$ is smaller than the Compton wavelength, with the difference that the separation into two distinct wave packets becomes manifest imediately, and not after certain period. Such exact initial localization (as a rectangular wave packet), of course, is not invariant under Lorentz transformations. When observed from another frame, the simultaneity hypersurface $\Sigma^{\prime}$ is no longer the same, and it intersection with the evolving wave packet does not give an exact localization on $\Sigma^{\prime}$, but a localization with an infinite tail. The exception, as we have seen above, is the limiting case when the width of the exact localization goes to zero and we approach the localization at a spatial point. Such, initially $\delta$-like localized, wave packet does not instantly evolve into a wave packet with infinite tail, but remains localized on the light cone.

7 On the causality violation of a relativistic wave packet

Inspecting the wave packet in Fig. 2 one observes that the probability density extends accross the light “cone”. Many authors have analysed such behaviour in view of a possible causality violation [16, 17, 18, 19, 20, 21, 5, 15]. However, causality would be violated if information could be sent faster than light. The fact that some part of the probability density arrives at a position $x$ earlier than light, by itself does not guarantee that information can also arrive quicker than light. With a single particle one cannot send information, because the position $x$ at which the particle will be detected is uncertain. One needs a modulated particle beam, e.g., a sequence of pulses of many particles, a statistical mixture of them. Then the sum $\sum_{i}|f_{i}|^{2}$ is proportional to the density $\rho$ of particles at a position $x$ at a time $t$. In Fig. 4 it is shown how the density at the fixed position $x=1$ changes with time in the situation in which after a first wave packet $f_{1}$, formed at $t=0$, a second, similar, wave packet $f_{2}$, formed at $t=0.1$, is emitted. In the right plot both densities are summed. We see that at $t<1$ there is no modulation of the particle density, which indicates that in such an arrangement information cannot be transmitted faster than light. The fact that $\rho$ starts to increase before $t=1$, which in this units is the arrival time of light, does not automatically imply that a message has been received at $t<1$, because at that earlier time there has been no obvious modulation of the density $\rho$.

Alternatively, a beam of particles can be modulated spatially, e.g., by an arrangement of slits, and so bear a message or a signal. A possible setup is shown in Fig. 5 in which the wave packet wavelength $\lambda$ is small enough, so that the packet can go through any of the slits ${\cal S}$ more or less undisturbed. An alternative arrangement is shown in Fig. 6, where $\lambda$ is great enough for diffraction and interference effects to occur, so that spherical wave packets emerge from the hole and then interfere on the arrangements of slits ${\cal S}$. If the width of the wave packet is smaller than the Compton wavelength, then the message comes to the detectors faster than light. Because a superluminal effect of the wave packet is effectively observable within the Compton wavelength $\lambda_{c}$, the arrangement of detectors ${\cal D}$ should be within a distance $L<\lambda_{c}$. In order to be able to send a message into the past, the arrangement ${\cal D}$ should move with an appropriate velocity (see Refs. [10]). Moreover, such a message would arrive into the very nearby past (within the time that takes light to travel the distance $L<\lambda_{c}$).

We see that by using relativistic wave packets, we apparently cannot violate causality on the macroscopic level, because the experimental setups of Figs. 4 and 5 have either difficult to achieve or contradictory constraints. A mere look at Fig. 2, in which the width of the wave packet (and hence its superluminal tail) is smaller than $\lambda_{c}\sim 10^{-15}m$, reveals that causality, in the sense of sending a signal into a reasonably remote past, cannot be so easily violated, if at all. A very ingenious experimental setup would be necessary for a macroscopic observer being able to invoke causality violating situations à la “grand father paradox” or its simpler versions in which the apparatus is destroyed before emitting a signal. Even then, causality would be restored within a proper quantum mechanical description of the situation  [38, 39, 40, 36].

The above reasoning indicates that the issue of causality violation of relativistic wave packets is not so straightforward as it is usually assumed. Also Fleming [10] and Wagner [20] have come to a similar conclusion. Ruijsenaars [22] has pointed out that the detection of ‘acausal events’ is vanishingly small under present laboratory conditions. Eckstein and Miller [21] observed that “causality brekdown” has a transient character which, according to our finding, does not automatically imply the possibility information transmission into the past. Karpov et al. [34] pointed out that the two complex components forming an initially localized wave packet move causally, with the velocity of light in the opposite directions. In their example the initial wave packet of a massless particle was localized within a rectangle, and afterwards it had the long tails that decayed with the distance $x$ according to $b/x$ for $b/x<<1$, where $b$ was the size of the localized wave packet. They wrote [34]:

[Such long tails] are precursors to the usual wave propagation. Although we may have instant interactions, these are not result of superluminal propagation, but of “preformed” structures.

Further, Antoniou et al. [35] considered a quantum electrodynamics case and demonstrated the appearance of nonlocal effects at the level of states. They showed that the expectation value of the electromagnetic field spreads causally, and that the classical measurements cannot detect the “acausal” effects of this non-locality.

In this connection let me point out that with waveguides one can arrange setups in which the group velocity of waves, the so called evanescent waves, is greater than the velocity of light (see e.g.,  [41, 42]). There has been a lot of discussion about whether or not such evanescent waves can transmit information faster that light. Many authors agree that in such cases the group velocity is not the velocity of information transmission, and that information travels slower than light. But Nimitz [41] has shown that signals in such arrangements are indeed superluminal, and yet they do not violate causality in the sense that the effect cannot precede the cause. This is so, because a signal has a finite duration. Therefore, in a typical setup in which an observer $A$ sends a superluminal pulse-like signal to a fast moving observer $B$, the pulse-like signal sent back from $B$ to $A$, because of the pulse’s finite width, cannot arrive into the past of $A$.

In the case of evanescent waves a faster than light group velocity does not automatically imply causality violation. We have seen that also the existence of superluminal tails in relativistic wave packets does not automatically imply the possibility of superluminal communication and thus causality violation. Moreover, in previous section we have demonstrated that if an effective width of a wave packet goes to zero, then the probability density approaches the exact localization on the light cone. A wave packet behaves apparently “acausally” only if its width $\Delta x$ is smaller than the Compton wavelength $\lambda_{c}$, but if $\Delta x$ goes to zero, then the “acausal” behavior disappears.

A deeper and more detailed thorough analysis has to be done, before we can say for sure that causality in the sense of “the grand father paradox” can be violated with relativistic wave packets. And if it is apparently violated, then we should seek how to remedy the situation, and not reject prematurely the concept of relativistic wave packets. We have seen that relativistic wave packets, either in momentum or in position space, are unavoidable ingredients of relativistic quantum field theory, and may represent various types of a particle’s localization, including point-like, rectangular, or Gaussian-like localization.

8 Conclusion

We have clarified the well known difficulties regarding localization of states in relativistic quantum mechanics and quantum field theories. For this purpose we proceeded step by step and thus more or less reviewed certain known facts and results of quantum field theory, which enabled us to avoid some loopholes and point to connections that have been usually overlooked in the treatments that considered only a part of the full story.

In quantum field theory the basis states of the Fock space are created by the action of field operators on a vacuum. In order to obtain a generic state, one has to superpose such basis states by means of a wave packet profile (wave function) which, in general, is complex valued. It satisfies the Schroedinger equation with the Hamilton operator, which is positive definite with respect to the so defined Fock space states. Only positive frequencies occur in the wave function, whilst the field operators, expanded in terms of creation and annihilation operators, contain both positive and negative frequencies. A complex valued wave function should not be confused with a non hermitian field (operator). Therefore, even in the case of a hermitian field (operator), the corresponding wave function can be complex.

On a wave function and basis states one can apply a suitable functional transformation [50, 27, 26], such that the state remains the same. So we can transform the Klein-Gordon wave function into a new wave function, called [26] the Newton-Wigner-Foldy wave function, whose absolute square gives the probability density, either in momentum or position space. Similarly, we can transform the Klein-Gordon creation operators into new operators that create eigenstates of the Newton-Wigner position operators, i.e., Newton-Wigner localized states.

The type of localization is determined by the shape of of the wave function. It can be (i) point-like localization, or (ii) localization in a finite region of space vanishing outside, or (iii) localization in a finite region decaying with power or exponential law, or (iv) “effective” localization like a Gaussian wave packet. Usually, by “localization” is understood the localization of the type (i) or (ii), but in this paper we use the word “localization” for the localization of the type (iii) or (iv) as well. The wave packets and the corresponding probability currents can be transformed from one to another Lorentz frame. We have found that nothing unusual happens with the wave packets when observed from different Lorentz frames. A state localized around a certain position, remains localized in another frame as well. This is consistent with the fact that if we observe a wave packet in its rest frame, then it behaves approximately as a non relativistic wave packet which can be localized. If we observe the same wave packet from a moving frame, it remains localized. The very existence of particle pulses in accelerators confirms that even fast moving particles can be localized. However, a state, initially localized according to (ii) in a frame $S$, is localized according to (iii) if observed from another Lorentz frame $S^{\prime}$. This is so, because simultaneity is not invariant and because the type (ii) localization at $t=t_{0}$ in the frame $S$ is only momentary, immediately switching at any later time $t>t_{0}$ to the type (iii) localization. A special case is type (i) localization at $t=t_{0}$, which is a limiting case of the type (iv) localization when the width $\Delta x$ of a Gaussian-like wave packet approaches to zero. We have demonstrated that at later times the probability density is given by $|f(t,x)|^{2}$ which approaches to $|f|^{2}=\frac{1}{2}\left(\delta(t-x)+\delta(t+x)\right)$ if $\Delta x$ goes to zero. In the limiting case of a point-like initial localization the particle is thus localized on the light cone at any later instant. Also when observed from another frame, the particle remains localized on the light cone. The initial point-like localization is Lorentz invariant.

Despite that the wave packets whose width is smaller than the Compton wave length leak outside the light cone, they cannot be used for faster than light communication between macroscopic observers and devices. A transfer of information cannot be done with a single wave packet, but requires, e.g., modulated in time sequences of wave packets, which move at most with the velocity of light. On the contrary, a spatially modulated bunch of particles, localized within their Compton wavelength, can bring a signal or a message with a superluminal velocity to a position within the Compton length from the source. But the Compton wavelength of elementary particles is around $10^{-15}$ m (for electron) or smaller, so that an observer, even if by an ingenious way could send information (message, signal) into the past, that past would be only about $10^{-23}$ s from his present, so that no causality paradox of the “grand father paradox” or similar, could take place. But even if by an ingenious technology, creation of apparently paradoxical situations were possible at the macroscopic level, there would remain a possibility to explain them [38, 39, 40, 36] within an appropriate quantum setup [43, 44, 45, 46, 47, 48, 49].

We conclude that the usual arguments against localized relativistic states can be circumvented. Such states naturally occur within quantum field theory and are not problematic at all. This sheds new light on the implications of the Reeh-Schlieder theorem [30], which is interpreted as implying that states (including single particle states) cannot be exactly localized in a finite region (see, e.g., [23]). Such a conclusion comes from the fact that one of the axioms of algebraic quantum field theory [29] is causality. However, as pointed out by G. Valente [28], one has to distinguish among different concepts of ‘causality’ used in the literature, and not all imply the possibility of information transmission. Moreover, Karpov et al. [34] and Antoniou et al. [35] have demonstrated that the classical measurement cannot detect the “acausal” effects of the wave packet quantum states. In the scenario that occurred in the Reeh-Schlieder theorem, the superluminal influence of a field in one spacetime region to a field in another region cannot be used for a controlled transmission of information. Therefore, the Reeh-Schlieder theorem does not imply that quantum states cannot be localized in a finite region. They can be localized, but their immediate spreading over all the space, cannot be used for a superluminal transmission of information.

Appendix A: The propagator

The scalar product of two states (63) at different times can be expressed as

[TABLE]

where $\omega_{{\bm{x}}}\equiv\sqrt{m^{2}-\nabla^{2}}$. Using the Schrödinger equation (35) and (48), we have

[TABLE]

[TABLE]

The initial and final wave packet profiles $f_{1,2}(0,{\bm{x}})$ or ${\tilde{f}}_{1,2}(0,{\bm{x}})$ are arbitrary. Let us consider two choices:

[TABLE]

Then we obtain

[TABLE]

[TABLE]

which is just the Green function (109).

[TABLE]

then

[TABLE]

[TABLE]

where we have used $[{\tilde{a}}({\bm{x}}^{\prime}),{\tilde{a}}^{\dagger}({\bm{x}})]=(1/(2\omega_{\bm{x}}))\delta({\bm{x}}^{\prime}-{\bm{x}})$. Because ${\tilde{a}}^{\dagger}({\bm{x}})\equiv\varphi^{+}(0,{\bm{x}})$, $\varphi^{+}(t,{\bm{x}})={\rm e}^{-i\omega_{\bm{x}}t}\varphi(0,{\bm{x}})$, we can write Eq. (122) in the form ($x\equiv(t,{\bm{x}})$)

[TABLE]

If we do not impose the condition $t^{\prime}>t$, then the right hand side of Eq. (123) can be written in terms of the time ordered product $\langle 0|T\varphi(x^{\prime})\varphi(x)|0\rangle$, which is the usual QFT propagator.

Both propagators, (120) and (122) (i.e., (123)), are special cases of the scalar product (116)

In the case (i), the initial and final wave packet profiles are localized according to (119). This is the localisation studied in this paper. The initial, and analogously the final, state are then of the form

[TABLE]

and the scalar product (116) gives (120), which can be written as

[TABLE]

where the Hamilton operator in the ${\bm{x}}$ representation is $\omega_{\bm{x}}=\sqrt{m^{2}-\nabla^{2}}$. Using (57), the same localized state $f(0,{\bm{x}})=\delta^{3}({\bm{x}}-{\bm{x}}_{0})$ can be expressed in terms of the functions ${\tilde{f}}(0,{\bm{x}})$ as

[TABLE]

In the case (ii), the initial wave packet (and analogously the final wave packet) is determined by (121), so that

[TABLE]

The scalar product (122) can then be written in the form

[TABLE]

We have thus two kinds of propagators, (125) and (128), one between the states $|{\bm{x}}\rangle$, $|{\bm{x}}^{\prime}\rangle$, and the other one between the states $|{\tilde{\bm{x}}}\rangle$, $|{\tilde{\bm{x}}}^{\prime}\rangle$, which are all particular cases of a generic single particle state

[TABLE]

for two different choices, (119) and (126), of the wave packet profiles.

Explicit expression for $G(t,{\bm{x}}^{\prime}:t,{\bm{x}})$ is given by the expression (111), or the corresponding three dimensional expression considered in Ref. [7], whilst the explicit expression for the propagator (128) is [51, 52]

[TABLE]

From the latter expression it follows that the amplitude for the transition between the events separated by a space-like interval does not vanish. This fact has been explored within the context of the Dirac field in Ref. [52], where it was argued that contrary to the common understanding conveyed in the modern literature, such effect may have observable macroscopic consequences.

Acknowledgement

This work has been supported by the Slovenian Research Agency.