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--> --> --> -->A.Covariant fields
To obtain a gauge-invariant theory, we define the covariant fields $ A(\omega) = \exp\left(-\frac{{\rm i}}{2}\omega^{\hat\alpha\hat\beta} S_{\hat\alpha\hat\beta}\right) \in SL(2,{\mathbb{C}}) $ | (1) |
The Lagrangian theory of the covariant field
$ D_{\hat\alpha}^{(\rho)} = e_{\hat\alpha}^{\mu}D_{\mu}^{(\rho)} = \hat\partial_{\hat\alpha}+\frac{{\rm i}}{2}\, \rho(S^{\hat\beta\, \cdot} _{\cdot\, \hat\gamma})\,\hat\Gamma^{\hat\gamma}_{\hat\alpha \hat\beta}\,, $ | (2) |
$ \hat\Gamma^{\hat\sigma}_{\hat\mu \hat\nu} = e_{\hat\mu}^{\alpha} e_{\hat\nu}^{\beta}(\hat e_{\gamma}^{\hat\sigma} \Gamma^{\gamma}_{\alpha \beta} -\hat e^{\hat\sigma}_{\beta, \alpha})\,, $ | (3) |
When
$ x\to x' = \phi_{{\frak g}(\xi)}(x) = x+\xi^a k_a(x) +... $ | (4) |
In general, the isometries may change the relative position of the local frames, thus affecting the physical interpretation. For this reason, we propose the theory of external symmetry [26], in which we introduce the combined transformations
$ \Lambda^{\hat\alpha\,\cdot}_{\cdot\,\hat\beta}[A_{\frak g}(x)] = \hat e_{\mu}^{\hat\alpha}[\phi_{\frak g}(x)]\frac{\partial \phi^{\mu}_{\frak g}(x)} {\partial x^{\nu}}\,e^{\nu}_{\hat\beta}(x)\,, $ | (5) |
$ ({A_{\frak g}},{\phi _{\frak g}}):\quad \begin{aligned}{e(x)}& \to &{e'(x')}& = &{e[{\phi _{\frak g}}(x)]{\mkern 1mu} ,}\\{\hat e(x)}& \to &{\hat e'(x')}& = &{\hat e[{\phi _{\frak g}}(x)]{\mkern 1mu} ,}\end{aligned}$ | (6) |
$ (A_{\frak g},\phi_{\frak g}):\quad \psi_{(\rho)}(x)\to\psi_{(\rho)}'(x') = \rho[A_{\frak g}(x)]\psi_{(\rho)}(x)\,, $ | (7) |
$ (T_{\frak g}^{(\rho)}\psi_{(\rho)})[\phi_{\frak g}(x)] = \rho[A_{\frak g}(x)]\psi_{(\rho)}(x)\,, $ | (8) |
We have shown that the pairs
For each Killing vector
$ \begin{aligned}[b]X^{(\rho)}_{a}& = {\rm i}{\partial_{\xi^a} T_{{\frak g}(\xi)}^{(\rho)}}_{|\xi = 0}\\ \;\;\;\;\;\;\;& = -{\rm i}k^{\mu}_{a}D_{\mu}^{(\rho)}+\frac{1}{2}\, k_{a\, \mu;\,\nu}\,e^{\mu}_{\hat\alpha}\,e^{\nu}_{\hat\beta}\, \rho(S^{\hat\alpha\hat\beta})\,,\end{aligned}$ | (9) |
Another advantage of the Lagrangian theory is the possibility of introducing the relativistic scalar product
In this framework, for any conserved operator X and field
$ C[X]\; \; \to\; \; {\cal{X}} = :\langle\psi,X\psi\rangle: $ | (10) |
2
B.Covariant fields on the de Sitter expanding universe
The $ \eta^5_{AB}z^A(x) z^B(x) = -\frac{1}{\omega^2}\,. $ | (11) |
The group of external symmetry,
$ L_{AB}^{5} = i\left[\eta_{AC}^5 z^{C}\partial_{B}- \eta_{BC}^5 z^{C} \partial_{A}\right] = -iK_{(AB)}^C\partial_C , $ | (12) |
$ \xi_{(AB)} \; \; \to\; \; k_{(AB)\mu} = -z_A(x)\stackrel{\leftrightarrow}{\partial}_{\mu}z_B(x) , $ | (13) |
Exploiting the special coset structure of the de Sitter manifold,
The hyperboloid equation is solved simply in the conformal chart
$ \begin{aligned}[b]z^0(x) &= -\frac{1}{2\omega^2 t_c}\left[1-\omega^2({t_c}^2 - \vec{x}^2)\right]\,, \\ z^i(x) &= -\frac{1}{\omega t_c}x^i \,, \\ z^4(x) &= -\frac{1}{2\omega^2 t_c}\left[1+\omega^2({t_c}^2 - \vec{x}^2)\right]\,,\end{aligned} $ | (14) |
$ {\rm d}s^{2} = \eta^5_{AB}{\rm d}z^A(x){\rm d}z^B(x) = \frac{1}{\omega^2 {t_c}^2}\left({{\rm d}t_c}^{2}-{\rm d}\vec{ x}\cdot {\rm d}\vec{x}\right)\,. $ | (15) |
$ M_+ \; :\; \; -\infty<t_c = -\frac{1}{\omega}{\rm e}^{-\omega t}<0 \,, $ | (16) |
$ M_- \; :\; \; \; \; \; \; \; 0<t_c = \frac{1}{\omega}{\rm e}^{\omega t}<\infty\,. $ | (17) |
$ {\rm d}s^2 = {\rm d}t^2-{\rm e}^{2\omega t} {\rm d}{\vec x}\cdot {\rm d}{\vec x}\,, $ | (18) |
The isometry group
$ H = \omega X_{(04)}^{(\rho)} = -i\omega(t_c\partial_{t_c}+x^i\partial_i) = i\partial_t-i\omega x^i\partial_i \,,\; \; \; \; $ | (19) |
$ P^i = \omega(X_{(4i)}^{(\rho)}-X_{(0i)}^{(\rho)}) = -i\partial_i\,, $ | (20) |
Respecting ad litteram the principles of the quantum theory, we assume that the quantum states of
Finally, we note that another attempt using de Sitter space-times considers only the unitary and irreducible linear reps. of the
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A.Klein-Gordon field
In the chart $ \left( \partial_t^2-{\rm e}^{-2\omega t}\Delta +3\omega \partial_t+m^2\right)\Phi(x) = 0\,, $ | (21) |
$ \begin{aligned}[b] \Phi(x) =& \Phi^{(+)}(x)+\Phi^{(-)}(x)\\ =& \int {\rm d}^3p \left[f_{\vec{p}}(x)a(\vec{p})+f_{\vec{p}}^*(x)b^{\dagger}(\vec{p})\right] \,, \end{aligned} $ | (22) |
$ \langle f_{\vec{p}},f_{\vec{p}'}\rangle_{KG} = -\langle f_{\vec{p}}^*,f_{\vec{ p}'}^*\rangle_{KG} = \delta^3(\vec{p}-\vec{p}')\,, $ | (23) |
$ \langle f_{\vec{p}},f_{\vec{p}'}^*\rangle_{KG} = 0\,, $ | (24) |
$ \langle f,f'\rangle_{KG} = {\rm i}\int {\rm d}^3x\, a(t)^3\, f^*(x) \stackrel{\leftrightarrow}{\partial_{t}} f'(x)\,. $ | (25) |
$f \in \left\{ \begin{aligned}&{{{\cal{H}}_ + } \subset {{\cal{K}}_ + }}&{{\rm{if}}}&{{{\langle f,f\rangle }_{KG}} > 0{\mkern 1mu} ,}\\&{{{\cal{H}}_0} \subset {{\cal{K}}_0}}&{{\rm{if}}}&{{{\langle f,f\rangle }_{KG}} = 0{\mkern 1mu} ,}\\&{{{\cal{H}}_ - } \subset {{\cal{K}}_ - }}&{{\rm{if}}}&{{{\langle f,f\rangle }_{KG}} < 0{\mkern 1mu} .}\end{aligned} \right.$ | (26) |
The fundamental mode functions can be expressed as
$ f_{\vec{p}}(t,\vec{x}) = \frac{{\rm e}^{{\rm i} \vec{x}\cdot \vec{p}}}{[2\pi a(t)]^{\frac{3}{2}}}{\cal{F}}_p(t)\,, $ | (27) |
$ \left[\frac{{\rm d}^2}{{\rm d}t^2}+\frac{p^2}{a(t)^2}+m^2-\frac{9}{4}\,\omega^2\right] {\cal{F}}_p(t) = 0\,, $ | (28) |
$ \left({\cal{F}}_p, {\cal{F}}_p\right)\equiv i\,{\cal{F}}_p^*(t)\stackrel{\leftrightarrow}{\partial}_{t}{\cal{F}}_p(t) = 1\,, $ | (29) |
The general solution of Eq. (28) can be derived easily in the chart
$ {\cal{F}}_p(t_c) = \ c_1\phi_p(t_c) + c_2\phi_p^*(t_c)\,,\quad \phi_p(t_c) = \frac{1}{\sqrt{\pi\omega}}\,K_{\nu}({\rm i}pt_c)\,, $ | (30) |
$\nu= \left\{ {\begin{aligned}{\sqrt {\frac{9}{4} - {\mu ^2}} }\quad {{\rm{for}}}\quad {\mu < \frac{3}{2}}\\{{\rm i}\kappa {\mkern 1mu} ,\kappa = \sqrt {{\mu ^2} - \frac{9}{4}} }\quad {{\rm{for}}}\quad {\mu > \frac{3}{2}}\end{aligned}} \right.{\mkern 1mu} , $ | (31) |
$ \left|c_1\right|^2-\left|c_2\right|^2 = 1\,. $ | (32) |
The most popular vacuum is the adiabatic Bunch-Davies type [21], with
$ K_{{\rm i}\kappa} ({\rm i}pt_c)\propto \frac{1}{\Gamma\left(\dfrac{1}{2}-{\rm i}\kappa\right)}\left(\frac{{\rm i}pt_c}{2}\right)^{-{\rm i}\kappa}-\frac{1}{\Gamma\left(\dfrac{1}{2}+{\rm i}\kappa\right)}\left(\frac{{\rm i}pt_c}{2}\right)^{{\rm i}\kappa}\,, $ | (33) |
To avoid these difficulties, we recently defined the r.f.v., separating the frequencies in the rest frames just as in special relativity [22]. Thus, we found that the rest energy,
$ M = \kappa\omega = \sqrt{m^2-\frac{9}{4}\omega^2}\,, \quad m>\frac{3}{2}\omega\,, $ | (34) |
$ c_1 = -{\rm i}\left(\frac{p}{2\omega}\right)^{-{\rm i}\kappa}\frac{{\rm e}^{\pi\kappa}}{\sqrt{{\rm e}^{2\pi\kappa}-1}}\,, $ | (35) |
$ c_2 = {\rm i}\left(\frac{p}{2\omega}\right)^{-{\rm i}\kappa}\frac{1}{\sqrt{{\rm e}^{2\pi\kappa}-1}}\,, $ | (36) |
$ {\cal{F}}_p(t_c) = \sqrt{\frac{\pi}{\omega}}\left(\frac{p}{2\omega}\right)^{-{\rm i}\kappa}\frac{I_{{\rm i}\kappa}({\rm i}pt_c)}{\sqrt{{\rm e}^{2\pi\kappa}-1}}\,. $ | (37) |
$ \mathop {\lim }\limits_{p \to 0} {\cal{F}}_p(t_c) = \frac{1}{\sqrt{2M}}\,{\rm e}^{-{\rm i} M t}\,, $ | (38) |
Finally, we must specify that the adiabatic and rest frame vacua are different from a physical perspective. First, they differ because in the adiabatic vacuum, the scalar particles may have any mass, whereas in the r.f.v. only the particles with
2
B.Dirac field
The Dirac field transforms under isometries according to a rep. induced by the Dirac rep. $ \rho_D(S^{\hat\alpha\hat\beta}) = \frac{{\rm i}}{4}\left[\gamma^{\hat\alpha}, \gamma^{\hat\beta}\right]\,. $ | (39) |
$ e_0 = \partial_t = \frac{1}{a(t_c)}\,\partial_{t_c}\,,\quad \; \; \; \; \omega^0 = {\rm d}t = a(t_c){\rm d}t_c\,, $ | (40) |
$ e_i = \frac{1}{a(t)}\,\partial_i = \frac{1}{a(t_c)}\,\partial_i\,, \quad \omega^i = a(t){\rm d}x^i = a(t_c){\rm d}x^i\,,\; \; \; \; \; $ | (41) |
$ D_x = {\rm i}\gamma^0\partial_{t}+ {\rm i} {\rm e}^{-\omega t}\gamma^i\partial_i +\frac{3{\rm i} \omega}{2} \gamma^{0}\,, $ | (42) |
The general solution of the Dirac equation may be written as a mode integral,
$ \begin{aligned}[b]\psi({x}\,)& = \psi^{(+)}({x}\,)+\psi^{(-)}({x}\,)\\ \; \; \; \; & = \int {\rm d}^{3}p \sum\limits_{\sigma}[U_{\vec{p},\sigma}(x){\frak a}(\vec{p},\sigma) +V_{\vec{p},\sigma}(x){\frak b}^{\dagger}(\vec{p},\sigma)]\,,\end{aligned} $ | (43) |
$ P^i U_{\vec{p},\sigma}(x) = p_iU_{\vec{p},\sigma}(x)\,, \quad P^i V_{\vec{p},\sigma}(x) = -p_iV_{\vec{p},\sigma}(x)\,, $ | (44) |
$ V_{\vec{p},\sigma}(t,\vec{x}) = U^c_{\vec{p},\sigma}(t,\vec{x}) = C\left[{U}_{\vec{p},\sigma}(t,\vec{x})\right]^* \,, \quad C = {\rm i}\gamma^2\,, $ | (45) |
$ \langle U_{\vec{p},\sigma}, U_{{\vec{p}\,}',\sigma'}\rangle_D = \langle V_{\vec{p},\sigma}, V_{{\vec{p}\,}',\sigma'}\rangle_D = \delta_{\sigma\sigma^{\prime}}\delta^{3}(\vec{p}-\vec{p}\,^{\prime})\; \; \; $ | (46) |
$ \langle U_{\vec{p},\sigma}, V_{{\vec{p}\,}',\sigma'}\rangle _D = \langle V_{\vec{p},\sigma}, U_{{\vec{p}\,}',\sigma'}\rangle_D = 0\,, $ | (47) |
$ \langle \psi, \psi'\rangle_D = \int {\rm d}^{3}x\, a(t)^{3}\bar{\psi}(x)\gamma^{0}\psi(x)\,, $ | (48) |
In the standard rep. of the Dirac matrices (with diagonal
$ U_{\vec{p},\sigma}(t,\vec{x}\,) = \frac{{\rm e}^{{\rm i}\vec{p}\cdot\vec{x}}}{[2\pi a(t)]^{\frac{3}{2}}}\left( \begin{array}{c} u^+_p(t) \, \xi_{\sigma}\\ u^-_p(t) \, \dfrac{{p}^i{\sigma}_i}{p}\,\xi_{\sigma} \end{array}\right)\,, $ | (49) |
$ V_{\vec{p},\sigma}(t,\vec{x}\,) = \frac{{\rm e}^{-{\rm i}\vec{p}\cdot\vec{x}}}{[2\pi a(t)]^{\frac{3}{2}}} \left( \begin{array}{c} v^+_p(t)\, \dfrac{{p}^i{\sigma}_i}{p}\,\eta_{\sigma}\\ v^-_p(t) \,\eta_{\sigma} \end{array}\right) \,, $ | (50) |
The time modulation functions
$ \left[{\rm i}\partial_{t_c}\mp m\, a(t_c)\right]u_p^{\pm}(t_c) = {p}\,u_p^{\mp}(t_c)\,, $ | (51) |
$ \left[{\rm i}\partial_{t_c} \mp m\, a(t_c)\right]v_p^{\pm}(t_c) = -{p}\,v_p^{\mp}(t_c)\,, $ | (52) |
$ v_p^{\pm} = \left[u_p^{\mp}\right]^*\,, $ | (53) |
$ |u_p^+|^2+|u_p^-|^2 = |v_p^+|^2+|v_p^-|^2 = 1 \\ $ | (54) |
$ u^{+}_p(t_c) = \sqrt{-\frac{p t_c}{\pi}}\left[c_1 K_{\nu_{-}}\left({\rm i} p t_c\right)+c_2 K_{\nu_{-}}\left(-{\rm i} p t_c\right)\right]\,,\; \; \; \; $ | (55) |
$ u^{-}_p(t_c) = \sqrt{-\frac{p t_c}{\pi}}\left[c_1 K_{\nu_{+}}\left({\rm i} p t_c\right)-c_2 K_{\nu_{+}}\left(-{\rm i} p t_c\right)\right]\,, $ | (56) |
$ |c_1|^2+|c_2|^2 = 1\,, $ | (57) |
Thus, we derive the general structure of the covariant Dirac field (43), for which the field operators
To completely determine our solutions we must adopt the criterion of frequency separation. The adiabatic vacuum may be defined simply by choosing
The solution is to adopt the r.f.v., imposing the conditions [23]
$ \mathop {\lim }\limits_{p \to 0} u_p^ - (t) = \mathop {\lim }\limits_{p \to 0} v_p^ + (t) = 0{\mkern 1mu} , $ | (58) |
$ c_1 = \frac{{\rm e}^{\pi\kappa}p^{-{\rm i}\kappa}}{\sqrt{1+{\rm e}^{2\pi \kappa}}}\,, \quad c_2 = \frac{{\rm i}\,p^{-{\rm i}\kappa}}{\sqrt{1+{\rm e}^{2\pi \kappa}}}\,, $ | (59) |
$ u_p^{\pm}(t_c) = \pm \frac{\sqrt{-\pi t_c}\, p^{\nu_-}}{\sqrt{1+{\rm e}^{2\pi\kappa}}}\, I_{\mp\nu_{\mp}}({\rm i}pt_c)\,. $ | (60) |
$ \mathop {\lim }\limits_{\vec p \to 0} {U_{\vec p,\sigma }}(t,\vec x) = \frac{{{{\rm e}^{ -{\rm i}mt}}}}{{{{[2\pi a(t)]}^{\frac{3}{2}}}}}\left( {\begin{array}{*{20}{c}}{{\xi _\sigma }}\\0\end{array}} \right){\mkern 1mu} , $ | (61) |
$ \mathop {\lim }\limits_{\vec p \to 0} {V_{\vec p,\sigma }}(t,\vec x) = \frac{{{{\rm e}^{{\rm i}mt}}}}{{{{[2\pi a(t)]}^{\frac{3}{2}}}}}\left( {\begin{array}{*{20}{c}}0\\{{\eta _\sigma }}\end{array}} \right){\mkern 1mu} , $ | (62) |
In contrast to the Klein-Gordon field, the r.f.v. of the Dirac field holds for any mass, separating the frequencies just as in special relativity. Nevertheless, its relation with the adiabatic vacuum is similar to that in the scalar case because the basis of the adiabatic vacuum is related to that of the r.f.v. through a non-trivial Bogolyubov transformation, the coefficients of which are proportional to the constants (59).
$ x = -p t_c = \frac{p}{\omega}{\rm e}^{-\omega t} $ | (63) |
2
A.Approximation method
We propose a generalization of the standard uniform asymptotic expansion (A8) to $ \begin{aligned}[b]J_{i\kappa+\lambda}(\kappa x) \simeq {\cal{J}}(\kappa,\lambda, x) =& \frac{{\rm e}^{\frac{\pi\kappa}{2}-\frac{{\rm i}\pi\lambda}{2}}}{\sqrt{2\kappa\pi}}\\ \; \; \; \; \;& \times\frac{{\rm e}^{{\rm i} \kappa\sqrt{1+x^2}-\frac{{\rm i}\pi}{4}}}{ (1+x^2)^{\frac{1}{4}}} \left(\frac{1}{x}+\sqrt{1+\frac{1}{x^2}}\right)^{-{\rm i}\kappa-\lambda}\,,\end{aligned} $ | (64) |
We start with the observation that, in our case, the variable (63) is positively defined without reaching the value
Figure1. (color online) Functions
$ {\cal{E}}(\kappa,\lambda,x) = 1-\frac{{|{\cal{J}}|}(\kappa,\lambda,x)|}{|J_{i\kappa+\lambda}(\kappa x)|}\,, $ | (65) |
Figure2. (color online) Function
The conclusion is that our approximation is numerically satisfactory even for
$ J_{{\rm i}\kappa+\lambda}(x)\simeq\frac{{\rm e}^{\frac{\pi\kappa}{2}-\frac{{\rm i}\pi\lambda}{2}}}{\sqrt{2\pi}}\,\frac{{\rm e}^{{\rm i} \sqrt{\kappa^2+x^2}-\frac{{\rm i}\pi}{4}}}{ (\kappa^2+x^2)^{\frac{1}{4}}} \left(\frac{\kappa}{x}+\sqrt{1+\frac{\kappa^2}{x^2}}\right)^{-{\rm i}\kappa-\lambda}\,, $ | (66) |
2
B.Flat limit of the Klein-Gordon field in r.f.v.
Let us briefly analyze the flat limits, for $ {\cal{F}}_p(t) = {\rm e}^{{\rm i}\delta_{KG}(p)}\left(\frac{p}{2\omega}\right)^{-{\rm i}\kappa}\sqrt{\frac{\pi}{\omega}}\frac{{\rm e}^{\frac{1}{2}\pi\kappa}}{\sqrt{{\rm e}^{2\pi\kappa}-1}}\,J_{{\rm i}\kappa}\left(\frac{p}{\omega}{\rm e}^{-\omega t}\right)\,. $ | (67) |
$ {\cal{F}}_p(t_c)\simeq\rho(p,t){\rm e}^{{\rm i}\theta(p,t)}\,, $ | (68) |
$ \rho(p,t) = \frac{1}{\sqrt{2}(M^2+p^2{\rm e}^{-2\omega t})^{\frac{1}{4}}}\frac{{\rm e}^{\pi\kappa}}{\sqrt{{\rm e}^{2\pi\kappa}-1}}\,, $ | (69) |
$ \begin{aligned}[b]\theta(p,t) = &\delta_{KG}(p)-\frac{\pi}{4}-\frac{M}{\omega} \ln\left(\frac{1}{2\omega}\right)-Mt\\ &-\frac{M}{\omega}\ln\left(M+\sqrt{M^2 +p^2{\rm e}^{-2\omega t}}\right)\\ &+\frac{1}{\omega}\sqrt{M^2+p^2{\rm e}^{-2\omega t}}\,.\end{aligned}$ | (70) |
$ \delta_{KG}(p) = \frac{\pi}{4}+\frac{M}{\omega}\ln \frac{M+\sqrt{M^2+p^2}}{2\omega}-\frac{\sqrt{M^2+p^2}}{\omega}\,. $ | (71) |
$ {\rm e}^{{\rm i}\delta_{KG}(p)} = {\rm e}^{\frac{{\rm i} \pi}{4}}\left(\frac{M+\sqrt{M^2+p^2}}{2\omega} \right)^{\frac{iM}{\omega}}{\rm e}^{-{\rm i}\frac{\sqrt{M^2+p^2}}{\omega}}\,, $ | (72) |
$ {\cal{F}}_p(t)\simeq\frac{{\rm e}^{\pi\kappa}}{\sqrt{{\rm e}^{2\pi\kappa}-1}}\frac{{\rm e}^{{\rm i}\theta_{KG}(p,t)}}{\sqrt{2}(M^2+p^2{\rm e}^{-2\omega t})^{\frac{1}{4}}}\,. $ | (73) |
$ \begin{aligned}[b] \theta_{KG}(p,t) =& -Mt -\frac{M}{\omega}\ln\left(\frac{M+\sqrt{M^2+p^2{\rm e}^{-2\omega t}}}{M+\sqrt{M^2+p^2}}\right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;&+\frac{1}{\omega}\left(\sqrt{M^2+p^2{\rm e}^{-2\omega t}}-\sqrt{M^2+p^2}\right)\,, \end{aligned}$ | (74) |
$ \begin{aligned}[b] \theta_{KG}(p,t) =& -\sqrt{M^2+p^2}\,t+\frac{\omega p^2 t^2}{2\sqrt{M^2+p^2}}\\ &-\frac{\omega^2 p^2(2M^2+p^2)t^3}{6(M^2+p^2)^\frac{3}{2}}+{\cal{O}}(\omega^3)\,, \end{aligned}$ | (75) |
$ \mathop {\lim }\limits_{\omega \to 0} {\cal{F}}_p(t) = \frac{{\rm e}^{-{\rm i}E(p)t}}{\sqrt{2E(p)}}\,, \quad E(p) = \sqrt{m^2+p^2}\,, $ | (76) |
2
C.Flat limit of the Dirac field in r.f.v.
To analyze how the flat limit can be reached in the case of the Dirac field, it is convenient to rewrite the time modulation functions (60) in the chart $\begin{aligned}[b] u_p^{\pm}(t) =& \pm {\rm e}^{{\rm i}\delta_D(p)\pm \frac{{\rm i}\pi}{4}} p^{-{\rm i}\kappa}\\ \;\;\;\;\;\;\;\;\; &\times \frac{{\rm e}^{\frac{\pi\kappa}{2}}}{\sqrt{{\rm e}^{2\pi \kappa}+1}}\sqrt{\frac{\pi}{\omega}\, p {\rm e}^{-\omega t}}\, J_{{\rm i}\kappa\mp\frac{1}{2}}\left(\frac{p}{\omega}{\rm e}^{-\omega t}\right)\,, \end{aligned}$ | (77) |
$ u_p^{\pm}(t)\simeq\rho^{\pm}(p,t) {\rm e}^{{\rm i}\theta(p,t)}\,, $ | (78) |
$ \rho^+(p,t) = \frac{{\rm e}^{\pi\kappa}}{\sqrt{{\rm e}^{2\pi \kappa}+1}}\,\frac{\sqrt{\sqrt{m^2+p^2{\rm e}^{-2\omega t}}+m}}{\sqrt{2}(m^2+p^2 {\rm e}^{-2\omega t})^{\frac{1}{4}}}\,, $ | (79) |
$\begin{aligned}[b]\rho^-(p,t) =& \frac{{\rm e}^{\pi\kappa}}{\sqrt{{\rm e}^{2\pi \kappa}+1}}\\ \; \;& \times\frac{p {\rm e}^{-\omega t}}{\sqrt{2}(m^2+p^2 {\rm e}^{-2\omega t})^{\frac{1}{4}}\sqrt{\sqrt{m^2+p^2{\rm e}^{-2\omega t}}+m}}\,,\; \; \; \;\end{aligned} $ | (80) |
$ \begin{aligned}[b] \theta(p,t) =& \delta_D(p)+\frac{\pi}{4}-mt\\ &-\frac{m}{\omega}\ln\left(m+\sqrt{m^2+p^2{\rm e}^{-2\omega t}}\right)\\ &+\frac{1}{\omega}\sqrt{m^2+p^2{\rm e}^{-2\omega t}}\,. \end{aligned} $ | (81) |
$ p{\rm e}^{-\omega t} = \sqrt{\sqrt{m^2+p^2{\rm e}^{-2\omega t}}+m}\,\sqrt{\sqrt{m^2+p^2{\rm e}^{-2\omega t}}-m} $ | (82) |
$ \delta_D(p) = -\frac{\pi}{4}+\frac{m}{\omega}\ln(\sqrt{m^2+p^2}+m)-\frac{1}{\omega}\sqrt{m^2+p^2}\,, $ | (83) |
$ u_p^{\pm}(t)\simeq \frac{{\rm e}^{\frac{\pi m}{\omega}}}{\sqrt{{\rm e}^{2 \frac{\pi m}{\omega}}+1}}\frac{\sqrt{\sqrt{m^2+p^2{\rm e}^{-2\omega t}}\pm m}}{\sqrt{2}(m^2+p^2 {\rm e}^{-2\omega t})^{\frac{1}{4}}}\, {\rm e}^{{\rm i}\theta_D(p,t)}\,. $ | (84) |
$ \begin{aligned}[b] \theta(p,t) =& -mt-\frac{m}{\omega}\ln\left(\frac{m+\sqrt m^2+p^2e^{-2\omega t}}{m+\sqrt{m^2+p^2}}\right)\\ &+\frac{1}{\omega}(\sqrt{m^2+p^2e^{-2\omega t}}-\sqrt{m^2+p^2}),\end{aligned} $ | (85) |
$ \begin{aligned}[b] \theta_{D}(p,t) =& -\sqrt{m^2+p^2}\,t+\frac{\omega p^2 t^2}{2\sqrt{m^2+p^2}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&-\frac{\omega^2 p^2(2m^2+p^2)t^3}{6(m^2+p^2)^\frac{3}{2}}+{\cal{O}}(\omega^3)\,, \end{aligned} $ | (86) |
Finally, we verify that, in the flat limit, we obtain the usual Minkowskian time modulation functions
$ \mathop {\lim }\limits_{\omega \to 0} u_p^{\pm}(t) = \sqrt{\frac{E(p)\pm m}{2 E(p)}}\, {\rm e}^{-{\rm i}E(p)t}\,. $ | (87) |
2
D.Physical consequences
Solving the problem of the flat limit, we derive suitable phases that complete the phases we introduced previously to define the r.f.v. We thus obtain the definitive form of the scalar time modulation functions, $ {\cal{F}}_p(t) = {\rm e}^{{\rm i}\alpha_{KG}(p)}\sqrt{\frac{\pi}{\omega}}\frac{{\rm e}^{\frac{1}{2}\pi\kappa}}{\sqrt{{\rm e}^{2\pi\kappa}-1}}\,J_{i\kappa}\left(\frac{p}{\omega}{\rm e}^{-\omega t}\right)\,, $ | (88) |
$ \begin{aligned}[b] \alpha_{KG}(p) =& \delta_{KG}(p)-\kappa \ln\left(\frac{p}{2\omega}\right) = \frac{\pi}{4}\\ \;\;\;\;\;\;\;\;\;\;\;&+\frac{M}{\omega}\ln \frac{M+\sqrt{M^2+p^2}}{p}-\frac{\sqrt{M^2+p^2}}{\omega}\,,\end{aligned} $ | (89) |
For the Dirac time modulation functions, we may write a similar result,
$ u_p^{\pm}(t) = \pm {\rm e}^{{\rm i}\alpha_D(p)\pm \frac{i\pi}{4}} \frac{{\rm e}^{\frac{\pi\kappa}{2}}}{\sqrt{{\rm e}^{2\pi \kappa}+1}}\sqrt{\frac{\pi}{\omega}\, p {\rm e}^{-\omega t}}\, J_{{\rm i}\kappa\mp\frac{1}{2}}\left(\frac{p}{\omega}{\rm e}^{-\omega t}\right)\,, $ | (90) |
$ \begin{aligned}[b]\alpha_D(p) = &\delta_{D}(p)-\kappa \ln p \\ = & -\frac{\pi}{4}+\frac{m}{\omega}\ln\left(\frac{\sqrt{m^2+p^2}+m}{p}\right)-\frac{\sqrt{m^2+p^2}}{\omega}\,, \end{aligned} $ | (91) |
These phases are so important in the de Sitter space-time because they do not affect the scalar products or contribute to the expressions of the transition probabilities. A specific feature of the de Sitter QFT is that the forms of some one-particle operators, including the energy one, are strongly dependent on the phases, which are functions of p. We remind the reader that, after canonical quantization, the one-particle energy operator is (19). In the case of the Klein-Gordon field, we consider the normalized mode functions (27) with the time modulation functions (88). Then, it is not difficult to verify the identity
$ (Hf_{\vec{p}}) = \left[-i\omega \left(p^i\partial_{p_i}+{\frac{3}{2}}\right)-\omega p^i\partial_{p^i}\alpha_{KG}(p)\right]f_{\vec{p}} , $ | (92) |
$ \begin{aligned}[b] {\cal{H}}_{KG} =& :\langle \Phi, H\Phi\rangle_{KG}: \\ \;\;\;\;\;\;\;\;= &\int {\rm d}^3 p\, \sqrt{M^2+p^2} \left[a^{\dagger}(\vec{p}) a(\vec{p}) +{b}^{\dagger}(\vec{p}){b}(\vec{p})\right]\\ \;\;\;\;\;\;\;\;&+\frac{i\omega}{2}\int {\rm d}^3p\, p^i \left\{ \left[\, a^{\dagger}(\vec{p})\stackrel{\leftrightarrow}{\partial}_{p_i} a(\vec{p})\right]\right.\\ & \left.+ \left[\, b^{\dagger}(\vec{p}) \stackrel{\leftrightarrow}{\partial}_{p_i} b(\vec{p})\right]\right\}\,, \end{aligned}$ | (93) |
$ \omega p^i\partial_{p^i}\alpha_{KG}(p) = -\sqrt{M^2+p^2}\,. $ | (94) |
$ \mathop {\lim }\limits_{\omega \to 0} {\cal{H}}_{KG} = \int {\rm d}^3 p\, \sqrt{m^2+p^2} \left[a^{\dagger}(\vec{p}) a(\vec{p}) +{b}^{\dagger}(\vec{p}){b}(\vec{p})\right]\,, $ | (95) |
$\begin{aligned}[b] K_{\nu}(z) = K_{-\nu}(z) = \frac{\pi}{2}\frac{I_{-\nu}(z)-I_{\nu}(z)}{\sin\pi \nu}\,, \end{aligned} \tag{A1} $ | (A1) |
$ \begin{aligned}[b] I_{\pm\nu}(z) &= {\rm e}^{\mp {\rm i}\pi\nu}I_{\pm\nu}(-z) \\ \;\;\;\;\;\;\;\;\;\;&= \frac{{\rm i}}{\pi}\left[K_{\nu}(-z)-{\rm e}^{\mp {\rm i}\pi\nu}K_{\nu}(z)\right]\,. \end{aligned} \tag{A2} $ | (A2) |
$ \begin{aligned}[b] {\rm i} I_{{\rm i}\kappa}({\rm i} s) \stackrel{\leftrightarrow}{\partial_{s}}I_{-{\rm i}\kappa}({\rm i}s) = \frac{2\, {\rm{sinh}}\,\pi\kappa}{\pi s}\,, \end{aligned}\tag{A3} $ | (A3) |
$ \begin{aligned}[b] {\rm i} K_{\nu}(-{\rm i} s) \stackrel{\leftrightarrow}{\partial_{s}}K_{\nu}({\rm i}s) = \frac{\pi}{|s|}\,, \end{aligned}\tag{A4} $ | (A4) |
For
$ \begin{aligned}[b] I_{\nu}(z) \to \sqrt{\frac{\pi}{2z}}{\rm e}^{z}\,, \quad K_{\nu}(z) \to K_{\frac{1}{2}}(z) = \sqrt{\frac{\pi}{2z}}{\rm e}^{-z}\,. \end{aligned}\tag{A5} $ | (A5) |
$ \begin{aligned}[b] I_{\nu}(z)\sim \frac{1}{\Gamma(\nu+1)} \left(\frac{z}{2}\right)^{\nu}\,, \end{aligned}\tag{A6} $ | (A6) |
The modified Bessel functions I are related to the usual ones as
$ \begin{aligned}[b] I_{\nu}(-ix) = {\rm e}^{-\frac{i\pi\nu}{2}}J_{\nu}\left(x\right)\,,\quad \forall\, x\in{\mathbb{R}},\, \nu\in{{\mathbb{C}}}\,, \end{aligned}\tag{A7}$ | (A7) |
$ \begin{aligned}[b] J_{i\kappa}(\kappa x) =& \frac{{\rm e}^{\frac{\pi\kappa}{2}}}{\sqrt{2\kappa}\,\pi}\,\frac{{\rm e}^{{\rm i} \kappa\sqrt{1+x^2}-\frac{{\rm i}\pi}{4}}}{ (1+x^2)^{\frac{1}{4}}} \left(\frac{1}{x}+\sqrt{1+\frac{1}{x^2}}\right)^{-{\rm i}\kappa}\\\; \; \; \; \; \;& \times\left[\sum\limits_{n = 0}^{n = N} \frac{(2 i)^n\Gamma(n+\frac{1}{2})}{\kappa^n (1+x^2)^{\frac{n}{2}}}\,a_n (x)+ {\cal{O}}(\kappa^{-N-1})\right]\,,\end{aligned} \tag{A8} $ | (A8) |
$\begin{aligned}[b] a_0(x) =& 1\,, \\ a_1(x)=& -\frac{1}{8}+\frac{5}{24}(1+x^2)^{-1}\,, \\a_2(x) =& \frac{3}{128}-\frac{77}{576}(1+x^2)^{-1}\\&+\frac{385}{3456}(1+x^2)^{-2},... \end{aligned}$ |