Arrival time in quantum field theory

Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.


Introduction
Free-motion time-of-arrival theories have been developed at the level of nonrelativistic quantum mechanics [1][2][3][4][5]. In Ref. [6] we have developed relativistic free-motion time-of-arrival theory for massive spin-1/2 particles. Relativistic quantum mechanics is only a transitional theory to quantum field theory, then for completeness, it is necessary to study arrival time at the level of quantum field theory. In traditional quantum field theory, energy and momentum are dynamical variables while time and space coordinates are parameters.
For our goal, our study is based on the event states satisfying the 4D spacetime interval μ ν = .
The first attempt to develop the field-quantized theory of time-of-arrival has been presented by A. D. Baute et al [7], in which the authors proposed a prescription for computing the density of arrivals of particles for multiparticle states both in free and interacting case, by applying the concept of the crossing state and the formalism of field quantization. Our work is different from theirs in the following aspects: (1) Their work is based on the nonrelativistic quantum theory, while our work is based on the relativistic quantum theory with a correct nonrelativistic limit; (2) Their work is only valid for the scalar particles, while our work is valid for both the spin-1/2 particles and the scalar particles by ignoring the spin degrees of freedom; (3) Their work is based on multiparticle states without a systematic quantum-field-theory framework, while our work is based on multievent states rather than multiparticle states, and a systematic quantum-field-theory framework is constructed via a position-momentum dual formalism.

Free-motion time-of-arrival operator of spin-1/2 particles and its eigenfunctions
To present a self-contained argument, let us first mention some of the contents presented in Ref. [6]. Let μ γ 's ( Here the 2D form is embedded in the 4D space-time background, and then the spin degrees of freedom should still be taken into account. The classical expression for the relativistic arrival time at the origin of the freely moving spin-1/2 particle having position x and uniform velocity There are many quantization schemes. For convenience, we choose Weyl's prescription as our quantization scheme, then the transition from the classical expression ) to a quantum-mechanical one requires us to symmetrize the product between x, E and p, and replace x, E and p with their quantum-mechanical operators, respectively, in such a way one can obtain the quantum-mechanical form of ( T x E p) = − [6,8]: is the nonrelativistic time-of-arrival operator that has been studied thoroughly in previous literatures [1][2][3][4][5], and it plays the role of proper time-of-arrival operator [6]. The operator given by Eq. (2) Assume that the eigenequation of one can prove that the eigenvalues and eigenfunctions of are, respectively (see where 1 1 0 For example, to examine whether Eq. (5) satisfies Eq. (4), one can apply the formulae: For our purpose, substituting p Here one can obtain Eqs. (11) and (12) (5) and (6), and then Eqs. (11) and (12) are completely equivalent to Eqs. (5) and (6), respectively. Just as a Hamiltonian operator such as ˆˆ( , ) H x p , the arrival time operator is a twovariable function of position and momentum. Likewise, is a twovariable function of x and p, where x and p are two independent variables such that x is not explicitly dependent on p and vice versa.
, in the following we take which has no effect on our final results.

Energy shift equation and proper-time eigenstates
Let us introduce an energy parameter ε with the dimension of energy and independent of the momentum (i.e., Eq. (13) is equivalent to Eq. (4), but it is expressed as a dual form of the Schrödinger However, the states given by Eqs. (14) and (15)  , which shows us a duality between the position and momentum space. To study quantum-field-theory arrival time we will resort to the position-momentum dual formalism, which is based on "event states" (proper-time eigenstates) instead of particle states (proper-mass eigenstates). Some previous attempts of describing event eigenstates can be found in Ref. [11][12][13], but they are based on the usual quantum theory, and different from our duality formalism with the following dual relations: In general, if a physical quantity Q satisfies 0 Q λ ∂ ∂ = for a parameter λ , we call Q a generalized conserved quantity with respect to the parameter λ (i.e., Q has a value constant in λ ). In particular, if λ is the energy parameter ε introduced before, it implies that the generalized conserved quantity Q is invariant under an energy shift, and then do not depend on the choice of zero-energy reference point. For example, it has no effect on a freely moving spin-1/2 particle to choose a new zero-energy reference point, because it is equivalent to putting the particle into a constant and uniform potential field. As one can define the Hamiltonian H. The Hamilton equations are In terms of the energy parameter ε , let us introduce a time function T as follows: Eq. (25) is the dual counterpart of Eq. (21). Using Eqs. (24) and (25), one has That is value. In the present case, using Eq. (28) we define a generalized Lagrange density as follows: In the two-dimensional form, the generalized Lagrange density Γ has the dimension of rather than that of In fact, when the time defined by a classical clock takes the value t, read the classical clock, of course the result is t without any fluctuation. However, in our case, the time takes the value T, read a quantum clock that follows the laws of quantum mechanics, and then the result has quantum fluctuations. Here the quantum clock that indicates its time as a function Dirac T of the distance is defined by the spin-1/2 particle and a "screen" (position-of-arrival).

Conclusions
Relativistic free-motion time-of-arrival operator for massive spin-1/2 particles, i.e., Dirac 1ˆˆˆ( , ) ( ) T x p x α βτ = − + , can be obtained via Weyl's quantization scheme [6]. To develop the free-motion time-of-arrival theory for massive spin-1/2 particles to the level of quantum field theory, we resort to a position-momentum dual formalism by means of event states