Lorentz-covariant spinor wave packet

For central position $X$ and momentum $P$ in $\left(d+1\right)$-dimensions, a Lorentz-invariant scalar wave packet $\left\lvert\Pi\right\rrangle$ is defined by [33, 34]:²²2See e.g. Refs. [30, 26] for reviews.

where $\Pi$ denotes the phase space³³3 Here, $\Pi$ includes the wave-packet central time $X^{0}$. Though $P^{0}=\sqrt{m^{2}+\boldsymbol{P}^{2}}$ is not an independent variable, we also include it for the convenience of writing its Lorentz transformation below.

and the normalization factor

provides $\left\llangle\Pi|\Pi\right\rrangle=1$, in which $K_{n}\!\left(z\right)$ is the modified Bessel function of the second kind. Here and hereafter, we fix $\sigma$ unless otherwise stated.

The wave function and the inner product are obtained as [33, 26]

where for any complex vector $V^{\mu}$, we write $\left\|V\right\|:=\sqrt{-V^{2}}$, namely,

with $\xi^{\mu}:=m\left[\sigma P^{\mu}+i\left(x-X\right)^{\mu}\right]$ and $\Xi^{\mu}:=m\left[\sigma\left(P+P^{\prime}\right)^{\mu}+i\left(X-X^{\prime}\right)^{\mu}\right]$.⁴⁴4 The abuse of notation is understood such that a vector-squared $V^{2}:=-\left(V^{0}\right)^{2}+\boldsymbol{V}^{2}$ is distinguished from the second component of $V$ by the context. We note that there is no branch-cut ambiguity for the square root as long as $m>0$  [26].

With this state, the momentum expectation value and its (co)variance become [33]

where

In general, a matrix element of $\hat{p}$ becomes

Let us consider a spacelike hyperplane $\Sigma_{N,T}=\Set{X}{N\cdot X+T=0}$ in the space of central position $X$; see Appendix A. One can write the completeness relation in the position-momentum phase space in a manifestly Lorentz-invariant fashion [26] (see also Ref. [33]):

where $\hat{1}$ denotes the identity operator in the one-particle subspace and the Lorentz-invariant phase-space volume element is given by

in which

is the Lorentz-covariant volume element. We stress that $\sigma$ is not summed nor integrated in the identity (22) and that the identity holds for any fixed $\sigma$.

Let us consider a “time-slice frame” $\check{X}$ of the central-position space in which $\check{\Sigma}_{\check{N},T}$ becomes an equal-time hyperplane $\check{X}^{0}=T$,

where the “standard” Lorentz transformation $\mathscr{L}\!\left(N\right)$ is defined by $N=:\mathscr{L}\!\left(N\right)\ell$, with $\ell$ denoting $\ell:=\left(1,\boldsymbol{0}\right)$ in any frame; note that $\check{N}=\mathscr{L}^{-1}\!\left(N\right)N=\ell$ by definition; see Appendix A for details. On the constant-$\check{X}^{0}$ hyperplane $\check{\Sigma}_{\check{N},T}=\Set{\check{X}}{\check{X}^{0}=T}$, the Lorentz-invariant phase-space volume element reduces to the familiar form:

Note that $\mathcal{M}_{\phi}\to 1$ in the non-relativistic limit $\sigma m^{2}\gg 1$.

In the literature [28, 29, 30, 31] a so to say spin-diagonal one-particle wave-packet state $\left\lvert\Pi,S\right\rrangle_{\text{D}}$ with a spin $S$ has been defined as

where $\left\llangle p\mathchoice{\;}{\;}{\,}{\,}\middle|\mathchoice{\;}{\;}{\,}{\,}\Pi\right\rrangle$ is nothing but the scalar Lorentz-invariant wave packet (11).⁵⁵5 In the literature, the normalization and $X$-dependence [26] have been omitted. Its normalization becomes

where $\left\llangle\Pi\mathchoice{\;}{\;}{\,}{\,}\middle|\mathchoice{\;}{\;}{\,}{\,}\Pi^{\prime}\right\rrangle$ is given in Eq. (15).⁶⁶6 An inner product of the spin-diagonal wave-packet state and another state $\Ket{\psi}$ is understood as $\left\llangle\Pi,S\mathchoice{\;}{\;}{\,}{\,}\middle|\mathchoice{\;}{\;}{\,}{\,}\psi\right\rangle_{\text{D}}:=\left(\left\lvert\Pi,S\right\rrangle_{\text{D}}\right)^{\dagger}\Ket{\psi}=:\left\llangle\Pi,S\right\rvert_{\text{D}}\Ket{\psi}$. We will never consider an inner product of the spin-diagonal wave-packet state and a phase-space-diagonal wave-packet state that appears below so that this notation will not cause confusion. This leads to the following completeness relation in the one-particle subspace

generalizing the completeness relation of the scalar wave packet (22).

Once the wave-packet state is defined, its Lorentz transformation law is obtained as

where

and

We see that the spin-diagonal choice (27) leads to the complicated transformation law (30) mixing the wave-packet state with the others having various centers of momentum and position.

Below, we will show that we can indeed realize a physically reasonable transformation law, so to say the phase-space-diagonal representation, which evades the mixing with other states (30):

where $C_{S^{\prime}S}\!\left(\Lambda,\Pi\right)$ is a yet unspecified representation function.

Instead of the conventional choice (27), we propose to define

where the key element is

and $N_{\psi}$ is a normalization factor to be fixed below; in the second step in Eq. (35), we used $u(p,s)=Cv^{*}(p,s)$, with the charge conjugation matrix $C=-\gamma^{2}$ ($=C^{*}=C^{\text{t}}=C^{\dagger}$) in our notation. Note that $M_{sS}\!\left(p,p\right)=\delta_{sS}$. Here and hereafter, for notational simplicity, we omit the label $N$ that distinguishes the particle and antiparticle unless otherwise stated.

The definition (35) leads to⁷⁷7 One can show it as $\displaystyle M_{sS}\!\left(p,P\right)$ $\displaystyle={1\over 2m}\overline{u}\!\left(p,s\right)S^{-1}\!\left(\Lambda\right)S\!\left(\Lambda\right)u\!\left(P,S\right)={1\over 2m}\sum_{s^{\prime},S^{\prime}}D^{*}_{s^{\prime}s}\bigl{(}{W\!\left(\Lambda,p\right)}\bigr{)}\overline{u}\!\left(\Lambda p,s^{\prime}\right)u\!\left(\Lambda P,S^{\prime}\right)D_{S^{\prime}S}\bigl{(}{W\!\left(\Lambda,P\right)}\bigr{)}$ $\displaystyle=\sum_{s^{\prime},S^{\prime}}D^{*}_{s^{\prime}s}\bigl{(}{W\!\left(\Lambda,p\right)}\bigr{)}M_{s^{\prime}S^{\prime}}\!\left(\Lambda p,\Lambda P\right)D_{S^{\prime}S}\bigl{(}{W\!\left(\Lambda,P\right)}\bigr{)}.$

Then it follows that

The identity (37) results in⁸⁸8 This can be shown as $\displaystyle\widehat{U}\!\left(\Lambda\right)\left\lvert\Pi,S\right\rrangle$ $\displaystyle=\sum_{s}\int{\text{d}^{d}\boldsymbol{p}\over 2p^{0}}\widehat{U}\!\left(\Lambda\right)\left\lvert p,s\right\rrangle\left\llangle p,s\mathchoice{\;}{\;}{\,}{\,}\middle|\mathchoice{\;}{\;}{\,}{\,}\Pi,S\right\rrangle=\sum_{s^{\prime},S^{\prime}}\int{\text{d}^{d}\boldsymbol{p}\over 2p^{0}}\left\lvert\Lambda p,s^{\prime}\right\rrangle\left\llangle\Lambda p,s^{\prime}\mathchoice{\;}{\;}{\,}{\,}\middle|\mathchoice{\;}{\;}{\,}{\,}\Lambda\Pi,S^{\prime}\right\rrangle D_{S^{\prime}S}\bigl{(}{W\!\left(\Lambda,P\right)}\bigr{)}$ $\displaystyle=\sum_{S^{\prime}}\left\lvert\Lambda\Pi,S^{\prime}\right\rrangle D_{S^{\prime}S}\bigl{(}{W\!\left(\Lambda,P\right)}\bigr{)},$ where we have used Eqs. (10) and (37) and then the unitarity (95) in the second equality.

As promised, we have realized the phase-space-diagonal representation (33).

Now we show that the normalization $\left\llangle\Pi,S\mathchoice{\;}{\;}{\,}{\,}\middle|\mathchoice{\;}{\;}{\,}{\,}\Pi,S\right\rrangle=1$ is realized by the choice

where $N_{\phi}$ is given in Eq. (13). Let us first compute

where we used Eq. (4) in the second line; we used $\left(\beta C\right)^{2}=-1$ and $\beta C\gamma^{\text{t}\mu}\beta C=\gamma^{\mu}$ in the last line, which is convenient for the antiparticle. The expectation value $\left\llangle\hat{p}^{\mu}\right\rrangle_{\phi}$ is presented in Eq (18), from which we get

Therefore, using the Dirac equation (3) and then the normalization (5), we see that the choice (39) provides the normalized state.

Finally, the inner product is given by the same procedure:

where, $\left\llangle-i\not{\hat{p}}\pm m\right\rrangle_{\Pi,\Pi^{\prime}}=-i\left\llangle\not{\hat{p}}\right\rrangle_{\Pi,\Pi^{\prime}}\pm m\left\llangle\Pi\mathchoice{\;}{\;}{\,}{\,}\middle|\mathchoice{\;}{\;}{\,}{\,}\Pi^{\prime}\right\rrangle$; see Eqs. (15) and (21). Hereafter, we adopt this representation for the Lorentz-covariant spinor wave packet.