A complete set of Lorentz-invariant wave packets and modified uncertainty relation

We define a set of fully Lorentz-invariant wave packets and show that it spans the corresponding one-particle Hilbert subspace, and hence the whole Fock space as well, with a manifestly Lorentz-invariant completeness relation (resolution of identity). The position-momentum uncertainty relation for this Lorentz-invariant wave packet deviates from the ordinary Heisenberg uncertainty principle, and reduces to it in the non-relativistic limit.


Introduction
Wave packets are one of the most fundamental building blocks of quantum field theory. We never observe a plane-wave state of, say, zero and infinite uncertainties of momentum and position, respectively. The plane-wave construction necessarily yields a square of the energymomentum delta function in the probability, which hence is always divergent and is more a mnemonic than a derivation (quoted from Sec. 3.4 in textbook [1]). However so far, it has been widely believed that there are no intrinsically new phenomena appearing from a wave-packet construction, but recent developments imply that it might play important roles in vast areas of science; see e.g. references in Introduction in Ref. [2].
Up to now, the wave-packet S-matrix has been computed using a complete basis of Gaussian wave-packets; see e.g. Refs. [3,4,5,2]. The Gaussian basis is constructed from a Gaussian wave packet that evolves in time t as e −i √ m 2 +p 2 t for each plane-wave mode p, and is not manifestly Lorentz covariant nor invariant. To fully exploit the Lorentz covariance of S-matrix in quantum field theory, it is desirable to have a complete basis of Lorentz-invariant wave packets. This is what we propose in this paper.
Here we stress a viewpoint that the Gaussian basis is equivalent to a complete set of coherent states in the position-momentum phase space (see Refs. [6,7,8,9,10] and references therein for works on coherent states in the context of the relativistic quantum mechanics 1 ). Guided by this equivalence, we develop the complete basis of Lorentz-invariant wave packets which is directly applicable in quantum field theory.
Our proposal is also inspired by the "relativistic Gaussian packet" [16,17] developed by Naumov and Naumov (see also Refs. [18,19,20]), and can be viewed as its generalization to form the complete set: From this viewpoint, our work can be interpreted as a new introduction of a spacetime center of wave packet as an independent variable, which is integrated over a spacelike hyperplane in the completeness relation along with a center of momentum.
It is worth mentioning that our Lorentz-invariant wave packet, when written in momentum space, is essentially the same as the one proposed in Refs. [6,7]. What is new in this paper in this respect is that we have also defined the wave function in position space and have computed it into an explicit closed form. Thanks to this, we can consider various limits to develop physical intuition. The momentum uncertainty we obtained is in agreement with that in Refs. [6,7], whereas the expectation value and uncertainty of the position on a constant time slice are obtained for the first time in this paper.
The organization of this paper is as follows: In Sec. 2, we review the plane-wave basis and the Gaussian basis, as well as the equivalence of the latter to the coherent basis, in order to spell out our notation. In Sec. 3, we present the Lorentz-invariant wave packet that we propose. In Sec. 4, we show the uncertainty relation on this state. In particular, we show that the position-momentum uncertainty deviates from that of the Heisenberg uncertainty principle, while the former reduces to the latter in the non-relativistic limit. In Sec. 5, we prove that these Lorentz invariant wave packets form a complete basis and that the completeness relation can be written in manifestly Lorentz-invariant fashion. As an example, we also show how a scalar field is expanded by this basis of Lorentz-invariant wave packets. In Appendix A, we show some of the known facts on the coherent states. In Appendix B, we present detailed computations for integrals that we encounter in the main text.

Gaussian basis and coherent states
Here in order to spell out our notation, we review basic known facts about the plane-wave basis and the Gaussian one, as well as the equivalence of the latter to the coherent one in the position-momentum space.

Plane-wave basis
We work in D = d + 1 dimensional flat spacetime with a metric convention (−, +, . . . , +) such that e ip·x = e −ip 0 x 0 +ip·x and p 2 = − p 0 2 + p 2 , where p 0 = −p 0 and a bold letter denotes a d-vector p = p 1 , · · · , p d = (p 1 , · · · , p d ), etc. Here and hereafter, x = x 0 , x are coordinates in an arbitrary reference frame. When p is on-shell, p 2 = −m 2 , p 0 = E p := p 2 + m 2 , and e ip·x = e −iEpx 0 +ip·x . Throughout this paper, we take the number of spatial dimensions d ≥ 2, all the momenta to be on-shell, and all the particles massive m > 0, unless otherwise stated. In particular, we use both of p 0 = E p (and of P 0 = E P appearing below) interchangeably.
In this paper, we focus on a free real scalar field that can be expanded in the Schrödinger picture as where a † p and a p are the creation and annihilation operators that obey where 1 is the identity operator on the whole Hilbert space, namely the Fock space H = ⊕ ∞ n=0 S L 2 R d ⊗n with S and L 2 R d being the symmetrization and the free one-particle momentum space, respectively. On this space, the free Hamiltonian H free can be expressed as up to a constant term. Similarly, the generator of the translation in the free theory is In the interaction picture, 2 where p 0 = E p = p 2 + m 2 as always.
We are focusing on the real scalar field in this paper because it is straightforward to generalize it to spinor and vector fields: We may expand these fields (in the interaction picture) as Strictly speaking, |x should be regarded as spanning the space K (x 0 ) of positive-energy solutions to the Klein-Gordon equation at x 0 , given the initial data L 2 Σ (0) on the Cauchy surface Σ (0) , whereas one would expect that this is equivalent to L 2 Σ (x 0 ) due to the timetranslational invariance of the theory. Hereafter, we write L 2 Σ (x 0 ) but a cautious reader may recast it into K (x 0 ) .
We can define the formal momentum operatorp on L 2 R d bŷ and the formal position operatorx as the generator of momentum translation on L 2 R d by 5 where (∇ p ) i = ∂ ∂p i . They satisfy the canonical commutator [x i ,p j ] = iδ ij1 , where1 has been the identity operator on the one-particle subspace L 2 R d as said above. Since we have chosen Σ (0) to be the Fourier transform of the momentum space R d , we also obtain on L 2 Σ (0) , which is consistent with Eqs. (10) and (13). We note thatx and |x are the time-independent ones in the Schrödinger picture by construction; recall footnote 2.
Here we stress that the position and momentum bases |x and |p have infinite norms x | x = ∞ and p | p = ∞, respectively, so that they do not belong to L 2 R d nor to L 2 Σ (0) . We never realize |x nor |p in any physical experiment. The formal position and momentum operatorsx andp and their eigenbases |x and |p , respectively, are mere mathematical tools to write down their expectation values basis-independently for any shape of normalizable wave packet |ψ in L 2 Σ (0) or L 2 R d as where This fact of the non-normalizability of basis is indeed one of the motivations of the Gaussian construction and its Lorentz-invariant generalization presented in this paper. The formal operatorx is first defined as the generator of momentum translation (13), and is associated with a particular foliation of spacetime through Eq. (14) such that Σ (0) is chosen as the Fourier transform of the momentum space R d via Eq. (10). Once this association with the position space is fixed,x is tied to the particular reference frame, with its unit spatial volume d d x manifestly violating the Lorentz invariance. The position operatorx is not covariant by construction; see also Ref. [15] for a review on the Lorentz non-covariance from the point of view of relativistic quantum mechanics.
We may regardp as a restriction of the momentum operator (4) to the free one-particle subspace L 2 R d : Schematically, Similarly, when restricted to the one-particle subspace, It also follows that, on L 2 Σ (0) , We may relate the bases of L 2 Σ (x 0 ) and L 2 Σ (0) by The position operatorx can be trivially extended to L 2 Σ (x 0 ) : where we used [x i , f (p)] = i ∂f ∂p i (p). On L 2 Σ (0) , the plane-wave normalization is 6 We may also formally write down the inner product of bases of L 2 Σ (x 0 ) and of L 2 Σ (x ′0 ) , The completeness relations on L 2 Σ (0) and on L 2 Σ (x 0 ) are, respectively, Mass dimensions are |x = |x = d 2 .
6 In Refs. [13,14] it has been shown that (what-we-call) "strict localization", which requires a wave function ψ(x) = x | ψ in L 2 Σ (0) to vanish everywhere outside a finite region V ⊂ Σ (0) , cannot be consistent with what the authors call "causality", which we will refer to the "Hegerfeldt causality". The Hegerfeldt causality holds when the following is satisfied: If ψ is strictly localized to V , then there should exist r that makes x| e −iĤ free x 0 +ip·a |ψ = 0 for all x ∈ V for all a with a > r at any later time x 0 > 0. The authors have proven that the Hegerfeldt causality is necessarily violated [13,14]. From Eq. (22), the position basis |x ′ , interpreted as a wave function of x in L 2 Σ (0) (closing our eyes on the fact that it cannot the case due to its non-normalizability), is strictly localized. Therefore, it obeys the proven violation of the Hegerfeldt causality. On the other hand, both the Gaussian wave packet (58) and our extension (104) have exponentially small but non-zero tail outside any finite region V from the beginning on Σ (0) and later. Therefore, they evade the condition of strict localization of the Hegerfeldt causality from the beginning.

Lorentz-friendly bases
From here we start to deviate from the standard notation in the literature. What-we-willcall "Lorentz-friendly basis" is essentially the same as the basis proposed by Newton and Wigner [11], which is later complemented in terms of the Euclidean group by Wightman [12].
We define a "Lorentz-friendly" annihilation operator on H: which gives and where the Lorentz invariance is made manifest in the last expression by letting p off-shell. We define a Lorentz-friendly momentum basis that spans L 2 R d : such that Mass dimensions are [α p ] = |p = − d−1 2 . This completeness is the same as Eq. (1) in Ref. [11] up to the factor 2 (which will not be mentioned hereafter).
We also define Lorentz-friendly position bases in L 2 Σ (0) and in L 2 Σ (x 0 ) , respectively: 7 Mass dimensions are |x = |x = d−1 2 . Here, |x and |x are generalizations of the one-particle position bases in the Schrödinger and interaction pictures, respectively. They satisfy 7 See footnote 4. Now we can write the field operator on H as Note that a wave function x | Φ is equivalent to the one given in Eq. (2) in Ref. [11]. On L 2 Σ (0) , we may formally write The normalization on L 2 Σ (0) is and we may again write down the inner product of bases of L 2 Σ (x 0 ) and of L 2 Σ (x ′0 ) : 8 On L 2 Σ (0) , the completeness relation becomeŝ which can be checked by sandwiching both-hand sides by p| and |p ′ . The same relation holds on L 2 Σ (x 0 ) when we replace |x and x| by |x and x| , respectively, because the factors e ±i √ m 2 +∇ 2 x 0 cancel out each other. Now we may rewrite the above completeness relation in a manifestly Lorentz-invariant fashion 9 on an arbitrary L 2 (Σ) space, with the same precaution as given after Eq. (11): 8 From this, the Feynman propagator is given by One may find its explicit form as a function of (x − x ′ ) 2 e.g. in Ref. [23]. 9 More precisely, the relation becomes Lorentz invariant when both-hand sides are sandwiched by the basis states p| and |q .
where d d Σ µ is the surface element normal to Σ. (In the language of differential forms, it is nothing but the induced volume element d d Σ µ = − ⋆dx µ , with ⋆ denoting the Hodge dual; in the flat spacetime, we get ⋆dx µ = 1 d! ǫ µµ 1 ...µ d dx µ 1 ∧ · · · ∧ dx µ d with ǫ 01...d = 1.) Physically, a probability density P (x) (per unit volume d d x) of observing the particle at a position x at time x 0 for a (normalized) wave packet |ψ is given by the expectation value of the projector |x x| on L 2 Σ (x 0 ) : 10 Note that the probability density is not mere an absolute-square of the wave function.
The following relations on L 2 R d may be useful: and we getx on L 2 Σ (0) and on L 2 Σ (x 0 ) , respectively. The relation (40) is equivalent to Eq. (11) in Ref. [11] up to the metric sign convention. We may formally define non-Hermitian position-like operator: 11 which satisfies on L 2 Σ (0) and on L 2 R d , respectively. Note that e.g. |x is not an eigenbasis ofχ but of χ † . In Eq. (44), the eigenvalues happen to be real for the non-Hermitian operatorsχ andχ † , respectively. We summarize the results for various bases in Table 1. 10 We are working in the interaction picture and hence the time dependence of the wave function is e i H free x 0 e −i Hx 0 |ψ with H = H free + Hint. Throughout this paper, we are neglecting interactions, Hint = 0, and hence it suffices to treat the time dependence as in the main text. 11 This operator has been discussed in Ref. [19] and references therein, whereχ is treated as self-adjoint. Our claim differs in thatχ is manifestly non-self-adjoint.
In each set of three rows separated by the horizontal lines, the first, second, and third rows are given in the Hilbert spaces L 2 R d , L 2 Σ (0) , and L 2 Σ (x 0 ) , respectively.
Finally, we comment on the Lorentz-transformation property of the one-particle momentum and position operators on L 2 R d or L 2 Σ (x 0 ) . The Poincaré transformation 12 on the annihilation operator reads (see e.g. Ref. [1]) where p Λ denotes the spatial component of Λp, namely (p Λ ) i = (Λp) i . In particular, We may reinterpret this transformation on states as that on operatorŝ The momentum operator is covariant on L 2 R d in this sense. The transformation (47) yields From Eqs. (31) and (51), we see that Again, reinterpreting this as transformation on operatorŝ we can show thatχ We see that, on the Lorentz-friendly basis |x in L 2 Σ (x 0 ) , the non-Hermitian operatorχ † is the spatial component of a Poincaré-covariant vector. On the other hand, we clearly see that the physical position operatorx =χ † − ip 2E 2 p is not a spatial component of a Lorentzcovariant vector due to the second term. From the view of modern quantum field theory, it is not compulsory thatx, associated to a particular spacetime foliation Σ (x 0 ) with time slices of constant x 0 , be a covariant operator. Of course, the whole theory is Lorentz invariant in the sense that the S-matrix, constructed from the covariant quantum fields (51) defined in the whole space H, is Lorentz invariant. 12 Here we only consider orthochronous Λ so that U free (Λ, b) is linear and unitary.

Gaussian basis
We define the Gaussian basis states through a normalizable, hence physical, wave function on where X and P are the centers of position and momentum of the wave packet, respectively, at time x 0 = 0 for Eq. (55) and x 0 = X 0 for (56), while σ > 0 is its width-squared. We see that these states on L 2 R d are related by |σ; X, P = e iEpX 0 |σ; X, P .
Due to this dependence on X 0 , one might want to regard the physical states |σ; X, P and |σ; X, P as some bases in the Schrödinger and interaction pictures on L 2 Σ (0) and L 2 Σ (X 0 ) , respectively, through the Fourier transformation. 13 However, when we consider the wave function (59) below, X 0 is rather a parameter that specifies the shape of the wave packet, and the time coordinate is x 0 . Again through the Fourier transformation, we may map the momentum-space wave functions onto L 2 Σ (0) and L 2 Σ (x 0 ) : x | σ; X, At the leading saddle-point approximation for large σ, Eq. (59) reduces to a closed form [4]: where P is on-shell P 0 = E P as always, and we define V := P /P 0 . We see that the center of wave packet moves as X + V x 0 − X 0 when we vary time x 0 , namely, when we change the time-slice Σ (x 0 ) . The inner product in L 2 R d is not orthogonal: where σ A := σ+σ ′ 2 and σ I : are the average and the inverse of average of inverse, respectively [5]. 13 See also footnotes 4 and 5.
It is important that the Gaussian basis, with any fixed σ, form an (over)complete set in the free one-particle subspace L 2 R d [3,4]: Because any fixed σ suffices to provide the complete set spanning L 2 R d , we omit the label σ unless otherwise stated hereafter. We may expand any wave function (or field configuration obeying the Klein-Gordon equation) ψ(x) = x | ψ by the Gaussian complete set { |X, P } X,P that spans L 2 R d : and conversely, the expansion coefficient may be computed by

Coherent states
Here we see that a Gaussian wave packet is indeed a coherent state [24,25] in the free-oneparticle subspace L 2 R d , or equivalently in L 2 Σ (0) . We define an "annihilation" operator for a d-dimensional harmonic oscillator: where λ is an overall normalization, which is usually taken to be λ = 1/ √ 2 but we leave it as an arbitrary complex number here. (More specifically,p has been defined on L 2 R d andx the generator of translation on it; or equivalently, one may regardx to be defined on L 2 Σ (0) andp the generator on it.) The dimensionality is given by [â] = [λ]. Note that this annihilation operator has nothing to do with the field annihilation operator in Eq. (2). We we get the solution in L 2 Σ (0) : where we have normalized such that α | α = 1 and hence [|α ] = 0. In the momentum space see Eq. (55). Physically, the real and imaginary parts of α correspond to the center of position and momentum of the Gaussian wave packet. Looking at Eqs. (68) and (69), it is rather mysterious why the wave functions take such particular forms as functions of complex numbers α. We will shed some light on this point in Sec. 3. Comparing Eq. (68) with Eq. (58), we see that the Gaussian wave-packet state is indeed a coherent state in L 2 R d or equivalently in L 2 Σ (0) : By taking λ = √ σ and −i/ √ σ, we may write Now we see that the completeness relation (62) is equivalent to the completeness of the coherent states in L 2 R d or equivalently in L 2 Σ (0) : where We list some more usable facts in Appendix A. We comment that the coherent state in the position-momentum space (68) or (69) should not be confused with the coherent state in the (photon) field space, used in quantum optics [24,25], for a fixed wavenumber vector k (and hence with the fixed wavelength 2π/ |k|): where a † k is the creation operator in the sense of Eq. (2) (but with a box normalization a k , a † k ′ = δ k,k ′ ) and we have taken λ = 1/ √ 2.

Lorentz-invariant wave packet
From the form of the Gaussian wave-packet state (56), it is tempting to generalize it into a Lorentz invariant form: where we have used the on-shell condition p 2 = P 2 = −m 2 . As we have seen in Eq. (71), the Gaussian wave-packet state |X, P is nothing but a position-momentum coherent state. For the coherent state, it is rather mysterious why the real and imaginary parts of the complex numbers α appear in the forms (68) and (69). It is remarkable that the Lorentz invariant generalization (75) has the seemingly holomorphic dependence on the D complex variables X + iσP if one generalizes P to be off-shell. 15 Motivated by this fact, we define the following Lorentz-invariant wave-packet state in where N σ is a normalization constant. Given the reference frame x, this wave packet is centered near x = X and p = P at time x 0 = X 0 . As said above, one might want to regard the state |σ; X, P as a basis of Σ (X 0 ) in the interaction picture but we will see that, in terms of the wave function (104) in Σ (x 0 ) , X 0 is mere a parameter that specifies the wave packet |σ; X, P , while the time is specified by x 0 . Also, we will continue to abbreviate σ to write |X, P unless otherwise stated.
As an illustration of more general computation spelled out in Appendix B, we will show in Sec. 3.1 that the normalization in L 2 R d , is realized by where K is the modified Bessel function of the second kind. With this normalization, mass dimensions are |X, P = 0 and [N σ ] = − d−1 2 . We comment on possible generalizations of P to be off-shell. If we make P off-shell in the first line in Eq. (76), it becomes divergent for |p| → ∞ when p · P > 0, namely, when where v := p/E p with |v| < 1. Therefore, the generalization of P to off-shell would be safe so long as P is timelike and future-oriented, in which case the condition (79) is never met. (This is the case too if we let P be off-shell in e −ip·X− σ 2 (p−P ) 2 = e −ip·X e σ 2 (m 2 −P 2 )+σp·P , though the limit of super-heavy "off-shell mass" −P 2 → ∞ diverges.) 15 As we will discuss below, the generalization of P to an off-shell momentum is straightforward so far as P is timelike and future-oriented. We leave further generalization for future study. 16 To be precise, the state |σ; X, P is Lorentz covariant and the wave function (76) is Lorentz invariant.

Normalization
We compute the norm on L 2 R d : where we let p be off-shell in the last line to make the Lorentz invariance manifest. As P is on-shell, we may always find a Lorentz transformation Λ(P ) to its rest frame P := (m, 0) such that ΛP = P . Then we change the integration variable to p := Λp. Using the Lorentz invariance of the integration measure etc. as well as Λ −1 p · P = p · ΛP = p · P = − p 0 m, we get where We see that the normalization |X, P 2 = 1 is realized by Eq. (78). One can also check that this result is consistent with the master formula (93) with Ξ → 2σP and hence Ξ = √ −Ξ 2 → 2σm. In the following, we list several limits for the reader's ease. First, and in the limits σm 2 → 0 and ∞, we get, respectively, Here one might find it curious that a plane-wave limit σ → ∞ is equivalent to a non-relativistic limit m → ∞, and a particle limit σ → 0 to an ultra-relativistic limit m → 0. The non-relativistic limit m → ∞ of the Lorentz-invariant wave packet (76) comes back to the Gaussian form (56) up to the factor √ 2m, where E p = m + p 2 2m + · · · , we have used the limit (83), and have neglected O m −2 terms in the last exponent in Eq. (84). In the ultra-relativistic limit m → 0, we get where cos θ X := p · X/ |p| |X| and cos θ P := p · P / |p| |P |. In this limit, the original Gaussian suppression is made weaker. Especially along the direction of P , cos θ P = 1, there is no suppression for a large momentum |p| → ∞. This is the main obstacle of having a Lorentz-invariant wave packet for a massless particle.

Inner product
Let us compute the inner product of two Lorentz invariant wave packets: σ; X, P | σ ′ ; X ′ , P ′ on L 2 R d . Here and hereafter, a prime symbol ′ never denotes a derivative.
Motivated by the coherent states in the position-momentum space (71), we define the following complex variables We see that For later use, we define a "norm" of an arbitrary complex D-vector Ξ: which is not necessarily positive nor even a real number. 18 Now we may write To compute this, it is convenient to define the master integral: where Ξ is a dimensionless complex D-vector and the D-vector u (= p/m) is on-shell and off-shell for the first and second integrals, respectively. In Appendix B, we present a detailed evaluation of the integral, and the result is which is valid when ℜΞ is timelike (ℜΞ) 2 < 0 and future-oriented ℜΞ 0 > 0. This also implies that with N σ is given in Eq. (78). The integral in the inner product (91) corresponds to that is, From we see that ℜΞ is always timelike. Therefore we may use the result (93): where in various notations, Especially when σ = σ ′ , we have where

Wave function
Let us compute the wave function for the Lorentz-invariant wave-packet state on L 2 Σ (x 0 ) : Comparing with Eq. (92), we see the following correspondence: Obviously (ℜΞ) 2 = −σ 2 m 4 < 0, and we may use Eq. (93): where The explicit form of the wave function (104) is one of our main results. This reduces to the earlier one in Refs. [16,17] when d = 3 in the X → 0 limit and may be interpreted as its spacetime translation by X. We note that there is no branch-cut ambiguity for the argument (105) as long as m > 0; see the last paragraph in Appendix B.1.
Hereafter, we examine various characteristics of the above wave function. Firstly, along the line x = X + P s corresponding to the particle trajectory, with s being a real parameter, we get Ξ = mP (σ + is) and hence Ξ = m 2 (σ + is). For a point sufficiently apart from X along this trajectory, namely for s → ±∞, we get We see that the wave function is not suppressed along the direction of P : There is no exponential suppression for |s| → ∞, while the apparent power suppression ∝ |s| −d for | x | X, P | 2 is merely due to the broadening of the width of Gaussian wave packet in d-spatial dimensions (for a normalized wave packet, the height of center becomes lower and lower when the width is more and more broadened), as we will soon see below.
Secondly, let us further consider a point slightly away from this trajectory, x = X +P s+ǫ, where ǫ is a small spacelike D-vector: ǫ 2 > 0. (Here, for each s, a point on the trajectory X + P s is specified, and we parametrize the spacelike hyperplane containing that point by ǫ.) Then in the limit |s| → ∞, where we have discarded O ǫ 2 and O ǫ 3 terms in the imaginary and real parts of the exponent, respectively, and O(ǫ) terms in other places. We observe the plane-wave behavior, e iP ·ǫ , and we obtain the Gaussian suppression factor: exp − σ 2(s 2 +σ 2 ) ǫ 2 + P m · ǫ 2 . It is noteworthy that the more we go along the trajectory x = X + sP (namely the larger the |s| is), the larger the spatial width-squared ∼ s 2 + σ 2 /σ of this Gaussian factor becomes. For a wave-packet scattering, we may parametrize each of the incoming waves, a, and of the outgoing ones, b, such that the scattering occurs (i.e. the wave packets overlap) around finite region |s a | ∼ |s b | < ∞. If the scattering occurs within a large time interval, the in and out asymptotic states are given by s a → −∞ and s b → ∞, respectively. In such a case, we may approximate an in-coming/out-going wave packet by the near plane wave (107) better and better, whereas they still interact as wave packets rather than plane waves.
Thirdly, for the plane-wave expansion with large σ, the argument becomes 19 where we have taken up to the order of leading non-trivial real part, and the wave function becomes The corresponding probability density is where we have used the completeness (37). In particular on the line x = X + P s, with s being a real parameter, the quadratic terms of s cancel out in the exponent in Eq. (109): As promised, we have confirmed that the wave function does not receive the Gaussian suppression along this particle trajectory. We stress that in this sense, the Lorentz-invariant wave packet is not localized in time, just as the Gaussian wave packet reviewed in Sec. 2.3. If we further take the non-relativistic limit in the exponent of the large-σ expansion (109), it becomes where P 0 = m + m 2 V 2 + · · · . Comparing with the Gaussian wave packet (60), we see that the extra suppression factor exp − (V · (x − X)) 2 2σ (113) 19 On the other hand, when we take the non-relativistic limit m → ∞ first, we get Ξ = σm 2 + appears from the Lorentz-invariant wave packet, and the center of the Lorentz-invariant wave packet departs from the particle trajectory X−V x 0 − X 0 of the Gaussian wave packet (60). Finally, in the particle/ultra-relativistic limit σm 2 → 0, we get where the argument goes to Ξ → m (x − X) 2 − 2iσP · (x − X). If we further take the relativistic limit m → 0, we get 20

Uncertainty relations
We show how the uncertainty relation changes for the Lorentz-invariant wave packet. We study the momentum and position uncertainties in the first two subsections and then discuss the uncertainty relation in the next. Lastly, we comment on the time-energy uncertainty.

Momentum (co)variance
We want to compute the momentum expectation value p µ and its (co)variance (recall that we have been taking all the momenta on-shell, and hencep 0 = Ep = m 2 +p 2 ): where for any operatorÔ, we write the expectation value with respect to |X, P as Ô := X, P |Ô |X, P .
Since we identify the Schrödinger, Heisenberg, and interaction pictures at x 0 = 0, the expectation value (117) corresponds to a measurement on the spacelike hyperplane Σ (0) . A measurement on a different time slice Σ (x 0 ) is given by (118) 20 On the other hand, first taking the particle limit σ → 0 is tricky due to the branch cut: When x is located at a spacelike distance from X, namely (x − X) 2 > 0, when timelike, (x − X) 2 < 0, and when lightlike (x − X) 2 = 0, As we are only considering free propagation of the waves,Ĥ =Ĥ free , this is the same as for our application. In particular whenÔ only contains momentum operators such that Ô ,Ĥ free = 0, the expectation value becomes time independent Ô = Ô I x 0 .
First, we write where u = p/m is the D-velocity with u 0 = √ 1 + u 2 . To compute the above, we take derivatives of the master integral (93): where Ξ is off-shell. Once this is obtained, we may substitute Ξ = 2σmP , which is "on-shell", Ξ = √ −Ξ 2 = 2σm 2 . From Eqs. (205) and (206) in Appendix B, we can readout where Ξ = 2σm 2 , and hence Note that contraction of Eq. (123) with the flat metric η µν gives η µν p µpν = −m 2 as it should, due to the Bessel identity (208). We see that for a fixed σ and m, the (co)variance (124) becomes larger and larger for |P | → ∞ due to the second term. Furthermore, even the off-diagonal covariance for µ = ν is non-zero. This is due to the fact that, with P = 0, the Lorentz-invariant wave packet is boosted and is not spherically symmetric in the momentum space, unlike the Gaussian wave packet (56). The above results agree with Eqs. (4.4) and (4.5) in Ref. [6]. From Eq. (122), we obtain where the equality holds in the plane-wave/non-relativistic limit Ξ = 2σm 2 → ∞. It is curious that the mass constructed from the expectation value of D-momentum p µ becomes larger than the "intrinsic" mass m, no matter whether the particle is at rest P = 0 or not. This fact has been pointed out in Ref. [17].
In the plane-wave/non-relativistic expansion for large Ξ = 2σm 2 , we get and hence, where the dots denote terms of order Ξ −3 . As a cross-check, we can derive from Eq. (127) that and see that there two coincide. We show the result of the plane-wave expansion with large σ in Eq. (124): That is, If we instead perform the non-relativistic expansion for large m in Eq. (124), we obtain Several comments are in order: The first term in Eq. (134) reproduces the momentum variance for the ordinary Gaussian wave packet, which is spherically symmetric ∝ δ ij . The second term shows that even the off-diagonal covariance for i = j is non-zero, due to the boost in the momentum space mentioned above. The first term in the energy variance (133) is also due to the boost, and is canceled out when we take the Lorentz invariant combination p 2 − p 2 (= −m 2 − p 2 ): By subtracting both-hand sides of Eq. (133) from those of Eq. (134) contracted with δ ij , we obtain As mentioned above, we see that the mass constructed from the expectation value of p µ is increased from the intrinsic mass m.

Position (co)variance
Now let us compute the expectation value x i and its covariance: First, where we write and we have used the following identity: 21 We have definedx as a time-independent Schrödinger-picture operator in a certain frame. Therefore the expectation value (138) should correspond to measurement in an equal-time slice in this frame, and hence the appearance of the non-covariant velocity u u 0 rather than the covariant one u m . As we identify the Schrödinger, Heisenberg, and interaction pictures at x 0 = 0, the expectation value (138) corresponds to the measurement on the spacelike hyperplane Σ (0) . If we instead consider the time-dependent operatorx I x 0 := e iĤ free x 0x e −iĤ free x 0 in the interaction picture, we obtain Second, we may similarly compute 21 This may be derived for a general (timelike) D-vector Ξ as and then putting the "on-shell" value Ξ = 2σmP . Recall that p and u are on-shell.
2u 0 , we can show that ℑ x ixj = 0. Then we obtain and hence 22 where we have used the identities (140) and In this paper, we compute the expectation value (138) and the (co)variance (144) using the saddle-point method for large σ: Especially for the variance i = j, where i is not summed. One may find the detailed derivation in Appendix B. Especially, we have used Eq. (255) to compute u i u 0 with Ξ = 2mσP and Ξ = 2σm 2 for x i , and similarly Eq. (256) for x ixj . The result (148) implies that if we measure the position uncertainty along the direction of P , it is Lorentz-contracted by the factor m/E P , compared to the measurement transverse to P [16,26,17]. 22 The last three terms in Eq. (144) may be recast into the form 1 we compute it as is.

Uncertainty relation
Finally combining Eqs. (132) and (148), the uncertainty relation on the time slice Σ (0) becomes where i is not summed. In the non-relativistic limit, we see that the terms of order P 2 /m 2 cancel out: where i is not summed. The ordinary minimum uncertainty for the Gaussian wave is recovered in the non-relativistic limit.
When we measure along a direction n with |n| = 1 and n · P = |P | cos θ, where we write A n := n·A for any spatial vector A. We see that the uncertainty is minimized to 1/2 when we measure along the directions θ = 0, π/2, and π; namely, either when it is parallel or perpendicular to P .

Time-energy uncertainty
Before proceeding, we comment on the time uncertainty. The Lorentz-invariant wave packet is not localized in time as discussed above, and therefore the expectation value of time "x 0 " for this wave packet is ill-defined, just as the expectation value of position is ill-defined for a plane wave. 23 However, we can show that the energy uncertainty (133) is matched with an uncertainty of the time at which this wave packet passes through a certain point x. Suppose we are at x = X and see the wave packet passing through it around the time x 0 ∼ X 0 . Then the probability density on each Σ (x 0 ) along the worldline x = X becomes From the exponent, we see that the timelike width-squared is (∆t) 2 ≃ σm 2 2P 2 . Comparing with the energy uncertainty (133), we see that the time-energy uncertainty takes the minimum value for the position-momentum one at the leading order: It would also be interesting to consider a wave-packet scattering. Then what is localized in time is not each wave packet but an overlap of the wave packets: This kind of timelike width-squared of the overlap region is given as σ t in Ref. [4] and as ς in , ς out in Refs. [5,2] for the Gaussian wave packets. This will be pursued in a separate publication.

Completeness of Lorentz-invariant basis
Let us discuss the completeness on L 2 R d . We will prove the following manifestly Lorentzinvariant completeness relation: where Σ X is an arbitrary spacelike hyperplane in the X space and d d Σ µ X is a d-volume element that is normal to Σ X . In the language of differential forms, (38) and below; see also Ref. [27] for discussion on Lorentz invariance of the phase space volume. In other words, where the right-hand side is the identity operator in the free one-particle subspace. Now the Lorentz-friendly plane wave is expanded as The completeness (154) can be rewritten as The proof of Eq. (157) is as follows: Noting that the left-hand side of Eq. (157) is manifestly Lorentz invariant (recall Eq. (27)), we may choose Σ to be a constant X 0 -plane without loss of generality: We may rewrite the completeness relation (155) in a different fashion: where To show this, we may sandwich both-hand sides by p| and |q , take the frame where Σ X becomes a constant-X 0 plane, and use Eq. (205). 24 The completeness (155) on the one-particle subspace L 2 R d can be naturally generalized to that on the whole Fock space H as follows. When we define A X,P by with mass dimensions A X,P = |X, P = 0, we obtain 25 and Now we get Putting Eq. (163) into the expansion (27), we obtain , we may obtain the explicit form of the above expansion coefficient: where Ξ = m (x − X − iσP ) 2 ; see Eqs. (103) and (105). The branch cut for the square-root in the argument is along that is, In a coordinate system x ′ that is a rest frame for P , the cut is along This is never satisfied and hence we are never on the cut; see also the last paragraph in Appendix B.1.
To cultivate some intuition, we show the case of Σ X being a constant-X 0 plane:

Summary and discussion
We have proposed a Lorentz-invariant generalization of the Gaussian wave packet. This Lorentz-invariant wave packet has more natural dependence on the central position X and momentum P than the coherent state in the position-momentum space: The dependence is holomorphic through the variable X + iσP if we further generalize it for a time-like off-shell momentum P . We have obtained the wave function for the Lorentz-invariant wave packet in an closed analytic form, as well as their inner products. The wave function is localized in space but not in time, while its width becomes larger and larger as the time is more and more apart from X 0 .
We have computed the expectation value and (co)variance of momentum for this state in a closed analytic form, and those of position in the saddle-point approximation. They reduce to the minimum position-momentum uncertainty of the corresponding Gaussian wave packet in the non-relativistic limit. The time-energy uncertainty takes the same minimum value at the leading order in the large width expansion.
We have managed to obtain the completeness relation for these Lorentz-invariant wave packets in a manifestly Lorentz-invariant fashion. It would be interesting to use the complete set of the Lorentz-invariant wave packets instead of the Gaussian ones in the decay and scattering processes analyzed so far. It would be worth applying to the wave packets in neutrino physics too. D †D =1, and thatD where we have defined The ground state of the harmonic oscillator |ϕ is given bŷ That is, |ϕ = |α α=0 , namely From the commutator we see thatâ iD (α) |ϕ = α iD (α) |ϕ , namely, and that the similarity transformation of the annihilation operator results in its displacement: In this sense, |α is the ground state of the harmonic oscillator displaced by α.

B Master integral
We encounter an integral of the form where p is on-shell andΞ is an arbitrary complex D-vector of mass dimension −1. For a massive particle p 2 = −m 2 < 0, it is more convenient to use the D-velocity u := p/m as an integration variable. Hereafter, we always take u "on-shell", u 2 = −1 and u 0 = √ 1 + u 2 , unless otherwise stated. Let us define where u is on-shell and off-shell for the first and second integrals, respectively, and Ξ is a dimensionless complex D-vector. Trivially substituting Ξ = mΞ, we get So far we have not put any kind of on-shell condition on Ξ, and hence where Ξ := √ −Ξ 2 = (Ξ 0 ) 2 − Ξ 2 as given in the main text. We write ℜΞ =: P and ℑΞ =: Q, which later will correspond to some momentum and position, respectively: Trivially, Ξ 2 = (P + iQ) 2 = P 2 + 2iP · Q − Q 2 .
On the other hand, we leave Q to be an arbitrary real D vector. We change the integration variable to u := Λu. Using the Lorentz invariance of the integration measure etc. as well as Λ −1 u · Ξ = u · (ΛΞ) = u · Ξ, we get Using u · Ξ = − u 0 Ξ 0 + u · Ξ = − u 0 Ξ 0 + i u · Q and renaming u by u, we get where Ω d−1 = 2π d 2 /Γ d 2 is the area of a unit (d − 1)-sphere (boundary of unit d-ball). We follow Ref. [23] in the following. The angular integral reads where J is the Bessel function of the first kind. (For d = 3, the right-hand side comes back to the familiar form 2 sin u Q /u Q .) Now where ε := √ 1 + u 2 is a rescaled energy. We use the second formula of Eq. (6.645) in Ref. [29]: where we used The limit Ξ → ∞ is whereas Ξ → 0 gives Recall that we are assuming d ≥ 2.
Throughout this paper, we choose to place a branch-cut for a square root, say √ z, on the negative real axis of z-plane: For −π < θ < π and r ≥ 0, √ re iθ := √ re iθ/2 .
In particular we may use the following limit for y → 0 under x > 0: Then for the expression (93), the condition on the argument to be on the real axis is As P 0 > 0, we see that Q 0 = 0 is it. However, the real part of the argument on the real axis ( Q 0 = 0) is positive: To summarize, no ambiguity arises from the branch cut as long as P is timelike: P 2 < 0. More in general, we may perform analytic continuation of the result (93) so long as ℑ −Ξ 2 ∝ P · Q = 0 or ℜ −Ξ 2 = Q 2 − P 2 > 0. On the other hand, possible non-triviality arises when Q 2 < P 2 (< 0) in the limit P · Q → 0: −Ξ 2 = −P 2 − 2iP · Q + Q 2 → −i sgn(P · Q) P 2 − Q 2 1 + i P · Q P 2 − Q 2 + · · · . (200)
Around the saddle point, we expand the integrand for large λ: e F = e F * + 1 2 ∆u i M ij ∆u j 1 + 1 3! ∂ 3 F * ∂u i ∂u j ∂u k ∆u i ∆u j ∆u k + 1 4! ∂ 4 F * ∂u i ∂u j ∂u k ∂u ℓ ∆u i ∆u j ∆u k ∆u ℓ where we have shifted the variables as ∆u := u − u * and have neglected terms of order 1/λ 2 , with ∆u being counted as 1/ √ λ. The derivatives at the saddle point are Now we may change the variables as with ∆u i = R ij ∆u j .