1 Introduction
For the discrete spectra of observables, of a finite-dimensional ket space with discrete eigenvalues, a complete set of eigenstates can be easily obtained by normalizing them. However, some observables, of an infinite-dimensional ket space, possess continuous eigenvalues in a continuous range. Among them, position and momentum are the typical examples, which have continuous eigenvalues. The eigenvectors of the continuum spectra are normalizable to delta function. Then naturally a question arises on how to make them a complete set of eigenstates.[1-4] Similar questions can be posed to the completeness relation of the photons in the free space and the electrons in the vacuum.
In order to answer the above questions, we study on the systems of quantum vacuum fluctuations, where the completeness relation of the photons in the free space has to be used. Firstly we consider the famous Casimir effect,[5-6] which demonstrates that the quantum vacuum fluctuation of the electromagnetic fields produces a macroscopic attractive force between the two closely paralleled plates.[7-9] Or, more generally, the zero point energy of the confined photon fields contributes to the force, which is $-\pi^2\hbar c/(240 a^4)$ with $a$ being the plate separation, arising when the virtual particles are excluded from the space between the plates as the separation $a$ is smaller than the wavelength of the particles. The Casimir force is precisely measured by Lamoreaux between parallel conducting plates.[10]
Recently, renewed attention has been focused on the Casimir effects branching out in various fields ranging from nanoscopic physics,[11] cold atomic physics[12-13] due to their unprecedented tunability and controllability in almost all aspects of the system parameters,[14-15] to solid state physics[16-18] and cavity system.[19]
Originally derived by using the quantum-mechanical perturbation theory to fourth order in $e$,[20] the Casimir force, in the standard approach, is obtained by computing the change in the zero-point energy per unit area of the electromagnetic field $E$ when the separation between perfectly paralleled conducting plates is changed, that is, $F_c=-\partial E/\partial a $. This derivation is mathematically much simpler. However, debate still exists. Schwinger had pointed out that the Casimir effect can be explained without reference to zero-point energies or even to the vacuum.[21] Also Jaffe realized that the concept of zero point fluctuations is not a necessity but a heuristic and calculational aid in the description of the Casimir effect.[22]
Moreover, in the calculation of the vacuum energy, infinite sums over the momenta lead to divergence, and therefore, artificial regulators are needed, to remove the divergence. For example, the zeta-function, heat kernel, and Gaussian regulator are introduced. However, after consideration on the completeness relation of continuum spectrum of photons, a spectrum density factor is naturally included to count for the number of eigenstates for the different momentum of the photons. As a result, the divergence is removed instead of the above mentioned artificial regulator.
With the help of the Casimir force, we settle down the explicit expression for the spectrum density factor $f(\omega)$ based on the completeness relation of the photons in the free space, that is, $f(\omega)={\rm e}^{-\sigma \omega}$, a Boltzmann distribution, where $\omega$ is the frequency of the electromagnetic wave between two surfaces, and $\sigma$, in unit of time, is a constant to be fixed by the experiments. The studies on the Casimir force show that quantum vacuum fluctuations have measurable consequences. For example, the atomic Casimir effect can account for the Lamb shift of spectra[23] and modify the magnetic moment of the electron.[24] Further application of the spectrum density factor $f(\omega)$ in the calculation of the Lamb shift fixes the undetermined constant $\sigma$. By further applying the spectrum density factor into the Lamb shift, we fix the undetermined $\sigma$. With that, we predict a small correction to the Casimir force.
The rest of the paper is organized as follows: in Sec. 2, we introduce a spectrum density of free photons in the free field or a statistic weight factor for photons of a momentum. In Sec. 3, we use this spectrum density to rederive the Casimir effect and settle down the explicit expression for the spectrum density factor. In addition, we obtain a small correction to the Casimir force. In Sec. 4, we fix the undetermined constant in the spectrum density factor.
Section 5 is devoted to conclusions and outlooks.
2 Spectrum Density Factor of Free Photons in the Free Field
We consider the typical Casimir effect with a pair of uncharged conducting metal plates at distance $a$ apart.
Assuming the parallel plates lie in the $xy$-plane, then, the virtual photons which constitute the vacuum field of quantum electrodynamics are free in $xy$ and confined in $z$ directions. The standing waves between the metal plates are
$ \psi_n(x,y,z;t) = {\rm e}^{-{\rm i}\omega_{k,n}t} {\rm e}^{{\rm i}k_xx+i k_yy} \sin \left(k_n z \right)\,, $
with $k_x, k_y\in (-\infty, \infty)$ the wave vectors free in the $xy$-direction. In this paper, we take the Dirichlet boundary condition (BC) in the $z$-direction: $\psi_n(x,y,0;t)=\psi_n(x,y,a;t)=0$. The Dirichlet BC requires
$$ k_n=\frac{n\pi}{a},~(n=1,2,\ldots)\,. $$
The frequency of the wave is
$ \omega_{k,n} = c\sqrt{k^2 + \frac{n^2\pi^2}{a^2}}\,, $
with $k^2=k_x^2 + k_y^2$. The vacuum energy (zero energy) per area is then,
$ E(a) = 2 \iint \frac{{\rm d}^2k}{(2\pi)^2} \sum_{n=1}^\infty \frac{\hbar}{2} \omega_{k,n}\,, $
where a factor of $2$ is responsible for the two possible polarizations of the wave. Here we notice that here are modes in the Neumann BC case that are not present in the Dirichlet BC case. But since their zero-point energy does not depend on the distance between the plates, they do not contribute to the Casimir force and can be discarded. From the expression of the vacuum energy, the Casimir force can be calculated,
$ F_c=\frac{\partial E(a)}{\partial a}\,. $
By noticing the summation over $n$ in Eq. (3), the result is clearly infinite! The computation of the Casimir force also leads to infinite sums, and therefore, requires regularization. The divergent sum for vacuum energy can be decomposed into an infinite and a finite part. Usually a regulator, such as, of zeta-function, heat kernel, or Gaussian, is introduced to make the expression finite, and in the end it will be removed without introducing extra effects. Normally the finite part does not depend on the choice of the regulator. Actually, Jared Kaplan has proofed in this course note[25] that, any regulator $R(x)$ in $E(a)$ gives
$ E(a) = \hbar\iint\frac{{\rm d}^2k}{(2\pi)^2} \sum_{n=1}^\infty R\Bigl(\frac{n}{a \Lambda}\Bigr) \omega_{k,n} \sim -\frac{\hbar c\pi^2 R(0)}{720 a^3}\,, $
and thus the correct attractive Casimir force as long as $ R(0) = 1 $. Here, $L$ is the constant of length dimension with $L \gg a$ and $\Lambda$ is the high momentum cutoff. Besides, two other requirements on $R(x)$ are that, the ultra high energy, short distance modes are irrelevant for the physics and the short distance regulator function $R(x)$ does not change the modes at very long distances, where the Casimir effect actually arises. At short distances the sub-leading terms at large $\Lambda$ are not exactly zero, and they do depend on the regulator function $R(x)$, which makes the regulator function $R(x)$ be observable. However, an artificially introduced regulator should not give extra effects.
To derive the Casimir force, it is not a necessity to induce the regularization. For example, it is possible to use the UV cut-off[26] or get rid of the vacuum energy.[27] In Casimir's original paper,[5] he compared the situation in which the plate is at a small distance and the situation in which it is at a very large distance. The difference between the two gives a finite attractive force.
In this paper, we present an alternative way to rederive the force by including a spectrum density factor without introducing a regulator (discussed in greater detail below and in the Appendix). Before going into the results, we have to discuss the completeness relation of the photons in the free space.[28]
For the photons in the free field, the eigenvector set of its momentum and energy is, $\{\vert { k}, \lambda\rangle^{(1)} \}$, where ${ k}$ is its wave vector (or momentum), and $\lambda$ its\linebreak two transverse polarizations. The eigenvector usually takes,
$ \vert { k}, \lambda\rangle^{(1)} \sim {\rm e}^{-{\rm i}{ k}\cdot { x}}\,. $
As shown in the Appendix, the eigenstates of the coordinate and momentum expanded in the Fock space are not unique, and usually, incomplete. For the photons in the free field, due to the nonconservation of the photon numbers, the eigenstate is thus not complete and also can not be used as a basis, and as a result, the physical quantities based on it lead to divergence. We thus transform an incomplete eigenvectors of the continuum observable into a unique complete set by introducing a spectrum density factor. Then we change Eq. (6) into
$ \vert { k}, \lambda\rangle = f({ k},\lambda) \vert { k}, \lambda\rangle^{(1)} = f({ k}) \vert { k}, \lambda\rangle^{(1)}\,, $
where $f({ k},\lambda)$ is the spectrum density factor. The second equality is due to the isotropic of the space. After the transformation, the set of eigenstates for photons in free space is given by Eq. (7) satisfying the completeness relation, and $f({ k})$ is the spectrum density or the statistic weight factor for photons of momentum $k$, to be determined by the relevant experiment, for instance, the experiments measuring the Casimir pressure between two gold-coated plates.[16] From the above analysis, we conclude that the number of eigenstates within ${ k}\rightarrow { k}+{\rm d}{ k}$ is $f({ k}) \vert { k}, \lambda\rangle^{(1)}{\rm d}{ k}$ but not simply the $\vert { k},\lambda\rangle^{(1)}{\rm d}{ k}$.
3 Casimir Effect
In this section, we will rederive the Casimir effect by considering the spectrum density factor for photons within momentum ${ k}+{\rm d}{ k}$. For the definite $n$, Eq. (2) gives,
$ k{\rm d}k=\frac{1}{c^2}\omega_{k,n} {\rm d}\omega_{k,n}\,, $
and Eq. (3) is rewritten as
$ E(a) = \hbar \sum_{n=1}^\infty \iint \frac{ k {\rm d}k {\rm d}\varphi}{(2\pi)^2} f(\omega_{k,n}) \omega_{k,n} \nonumber\\ =\frac{\hbar }{2\pi c^2} \sum_{n=1}^\infty \int_{cn\pi/a}^{\infty} f(\omega_{k,n}) (\omega_{k,n})^2 {\rm d}\omega_{k,n}\,. $
As we discussed in Sec. 2, a spectrum density factor $f(\omega_{k,n})$ is included in the above equation. Now we derive the analytic expression of $E(a)$ by assuming $f(\omega)={\rm e}^{-\sigma \omega}$, a Boltzmann distribution, where $\sigma$ has the unit of time, and is a constant to be determined by the experiments. Other forms of $f(\omega)$, for example, $f(\omega)={\rm e}^{-\sigma^2 \omega^2}$, are tested without leading to the correct Casimir force. We also want to emphasize that, the scheme we proposed here, is obviously different from the normally adopted regulation procedure, where the infinitesimal number is chosen to be $\sigma\rightarrow 0$ at the end of the calculation, while here, it is a undetermined number to be fixed.
$ E(a) = \frac{\hbar }{2\pi c^2} \frac{{\rm d}^2}{{\rm d}\sigma^2} \sum_{n=1}^\infty \int_{cn\pi/a}^{\infty}{\rm e}^{-\sigma\omega_{k,n}} {\rm d}\omega_{k,n} \nonumber\\ =\frac{\hbar }{2\pi c^2} \frac{{\rm d}^2}{{\rm d}\sigma^2} \sum_{n=1}^\infty \frac{1}{\sigma}{\rm e}^{-{cn\pi \sigma}/{a}} \nonumber\\ =\frac{\hbar }{2\pi c^2} \frac{{\rm d}^2}{{\rm d}\sigma^2} \frac{1}{\sigma} \Bigl[\frac{1}{1-{\rm e}^{-c\pi\sigma/a}}-1\Bigr]\,. $
Making use of the expansion,
$ \frac{1}{1-{\rm e}^x}=-\sum_{n=0}^\infty B_n \frac{x^{n-1}}{n!}~,~|x|\in (0, 2\pi)\,, $
where $B_n$ is the Bernoulli numbers,[29] we obtain the energy
$ E(a) = 3B_0 \frac{\hbar a}{\pi^2 c^3\sigma^4}-(1+B_1)\frac{\hbar}{\pi c^2\sigma^3}+B_4\frac{\pi^2\hbar c}{24 a^3} \\ \quad-B_5 \frac{\pi^3 \hbar c^2\sigma}{40 a^4}+B_6\frac{\pi^4\hbar c^3\sigma^2}{120 a^5}+\cdots $
Here, $B_0=1, B_1=-{1}/{2}, B_2={1}/{6}, B_3=0, B_4={1}/{30}, B_5=0$, and $B_6=-{1}/{42}$.
When the distance between the plates goes to infinity $a\rightarrow \infty$, it is the energy of the real vacuum,
$$ {\rm \lim_{a\rightarrow \infty}} E(a)= 3B_0 \frac{\hbar a}{\pi^2 c^3\sigma^4}-(1+B_1)\frac{\hbar}{\pi c^2\sigma^3}\,. $$
When choosing the zero energy to be the real vacuum, we obtain the effective energy per area between the plates
$ E_{\rm eff}(a)=B_4\frac{\pi^2\hbar c}{24 a^3}-B_5 \frac{\pi^3 \hbar c^2\sigma}{40 a^4}+B_6\frac{\pi^4\hbar c^3\sigma^2}{120 a^5}+\cdots\,, $
and the force per area
$ F_c=-\frac{\partial}{\partial a }E(a) =-\frac{\pi^2\hbar c}{240 a^4} -\frac{\pi^4 \hbar c^3\sigma^2}{1008 a^6}+\cdots $
It is seen that the plates do affect the virtual photons which constitute the field, and generate a net attractive force. The first term gives the usual attractive Casimir force. The second term is a small correction to the force, scaling as $\sigma \ll a/c$, and in turn can be used to fix the undetermined constant $\sigma$ in the spectrum density by the precise measurement on the force in the experiment. In all, $\sigma$ is a small constant number, which is consistent with our observation in that, only for photons with very high frequency, we can find the different behaviors through the spectrum density factor $f(\omega_{k,n})={\rm e}^{-\sigma \omega_{k,n}}$ we adopted, while in the normal circumstances where the photo frequency is not so high, the photons, contribute almost equally even with different frequencies since the spectrum density factor scales $f(\omega_{k,n})\sim 1$.
In the following section, we will try to fix the undetermined constant $\sigma$ through calculating the Lamb shift in the hydrogen atom.
4 Spectrum Density Factor in Lamb Shift
Lamb shift was first measured in 1947 by Lamb and Retherford to determine the splitting between the $2S_{1\over 2}$ and $2P_{1\over 2}$ states in Hydrogen to have a frequency of 1.06 GHz, which was not predicted by the Dirac equation.[30] The interaction between vacuum energy fluctuations and the hydrogen electron in these different orbitals accounts for the shift. This shift is now accurately measured to be 1057.864 MHz about the same size as the hyperfine splitting of the ground state.
Here we follow Welton's calculation by considering the fluctuation in the electric and magnetic fields associated with the quantum electrodynamic vacuum which perturbs\linebreak the electric potential due to the atomic nucleus. This perturbation causes a fluctuation in the position of the electron.[31] The difference of potential energy, which accounts for the Lamb shift is given by
$ \Delta V({ r}) =V({ r}+\delta{ r}) -V({ r}) \nonumber \\ = \delta{ r}\cdot \nabla V({ r})+\frac{1}{2}(\delta{ r}\cdot\nabla)^2V({ r})+\cdots\,, $
where the Coulomb potential is $V({ r})=-e^2/(4\pi\epsilon_0r)$. Considering the isotropic nature of the fluctuation $\langle\delta{ r}\rangle_{\rm vac}=0$ and $\langle(\delta{ r}\cdot\nabla)^2\rangle_{\rm vac}=({1}/{3})\langle (\delta{ r})^2\rangle_{\rm vac}\nabla^2$, one can obtain
$ \langle\Delta V\rangle =\frac{1}{6}\langle(\delta{ r})^2\rangle_{\rm vac}\left\langle \nabla^2\Bigl(-\frac{e^2}{4\pi\epsilon_0r} \Bigr)\right\rangle_{\rm at}, $
where $\langle...\rangle_{\rm vac}$ and $\langle...\rangle_{\rm at}$ denote the expectation on the vacuum fluctuation and the atomic orbitals, respectively. For the atomic orbital part,
$ \left\langle{\nabla^2\Bigl(-\frac{e^2}{4\pi\epsilon_0r}\Bigr)}\right\rangle_{\rm at}= -\frac{e^2}{4\pi\epsilon_0} \int {\rm d}{ r}\psi^{\ast}({ r})\nabla^2\Bigl(\frac{1}{r}\Bigr)\psi({ r}) \\ =\frac{e^2}{\epsilon_0}|\psi_{2S}(0)|^2=\frac{e^2}{8\pi\epsilon_0a_0^3}\,, $
where $\psi_{2S}(0) =1/(8\pi a^3_0)^{1/2}$, and $a_0$ is the Bohr radius $a_0=4\pi\epsilon_0\hbar^2/(me^2)$. The part of the mean square fluctuation is calculated as,[31]
$ {(\delta{ r})^2}_{\rm vac} =\frac{1}{2\epsilon_0\pi^2} \Bigl(\frac{e^2}{\hbar c}\Bigr)\Bigl(\frac{\hbar}{mc}\Bigr)^2\int^\kappa_{k_0} \frac{{\rm d}k}{k}\,, $
where the divergent integration is remedied by considering the upper limit and the lower limit of the wave vector. In the presence of any sort of binding, like the electron in the hydrogen atom, the lower limit $k_0$ will be determined from consideration of the details of the electronic motion. The electron is unable to respond to the fluctuating electromagnetic field if the fluctuations are smaller than the natural orbital frequency in the atom, which is $\pi/a_0$. The upper limit is taken by the limitation that the wavelengths must be longer than the Compton wavelength, or equivalently $k < mc/\hbar$. As a result, the difference of the potential energy becomes
$ {\nabla V}=\frac{\alpha^5mc^2}{6\pi}\ln\frac{1}{\pi \alpha}\,, $
where $\alpha ={({1}/{4\pi \varepsilon _{0}})}{({e^{2}}/{\hbar c})}$ is the fine-structure constant.
Historically the measurement for the Lamb shift and the theoretical development provided the stimulus for renormalization theory to handle the divergences. Here we use the spectrum density factor based on the completeness of the eigenvector set for the photon field $f(\omega)={\rm e}^{-\sigma \omega}$, that is, $f(k)={\rm e}^{-\sigma c k}$, to remove the divergence by a natural upper cutoff without introducing the upper limit, which means,
$ {(\delta{ r})^2}_{\rm vac} =\frac{1}{2\epsilon_0\pi^2} \Bigl(\frac{e^2}{\hbar c}\Bigr)\Bigl(\frac{\hbar}{mc}\Bigr)^2 \int^\infty_{k_0}\frac{{\rm d}k}{k}f(k)\,. $
Combined with Eq. (17), the Lamb shift becomes
$ {\nabla V}=-\frac{\alpha^5mc^2}{6\pi}{\rm Ei}(-\sigma c k_0)\,, $
where Ei is the exponential integral function. Comparing with the experimental results $1057.864$ MHz, we numerically obtain the constant $\sigma=1.37\times 10^{-23}$ s.
With this value, we estimate the correction of the Casimir force due to the second term in Eq. (14) to be $3.57\%$ of the first term with $a\sim 1~\mu$m. Recently, the Casimir pressure between two gold-coated plates was measured at $d =160$\;nm[16] and more elegant measurement was done between a metalized sphere and flat plate with the plate-sphere surface separations from $0.1\;\mu$m to $0.9\;\mu$m.[17] Casimir effects are also studied in a cold atomic sample of dilute Rydberg atoms trapped in front of a rough substrate.[12] We hope that the excellent experimental setup can be used to detect the correction based on our theory of spectrum density factor.
5 Conclusions
We have rederived the Casimir force between the two paralleled metallic plates by using the completeness of the eigenvector set for the photons in the free field. We have found that without introducing a regulator or the UV-cutoff, we obtained the Casimir force while including a spectrum density factor $f(\omega)={\rm e}^{-\sigma \omega}$. The first term we obtained gives the correct attractive Casimir force, $-\pi^2\hbar c/(240 a^4)$, and the second term we predicted, to be, $-\pi^4 \hbar c^3 \sigma^2/(1008 a^6)$.
By further applying the spectrum density factor into the energy difference induced by the interaction of the electron and the vacuum fluctuation, we compare our theoretical calculation with the experimental result, we settle down the undermined constant $\sigma$ in the spectrum density factor, which is $\sigma=1.37\times 10^{-23} $. By this constant, we give an estimation to the correction of the second term in the Casimir force to be $3.57\%$ relative to the first one with the distance between the two plates $a\sim 1\;\mu$m. The small difference between two could be determined by the precise experiments measuring the Casimir pressure between two gold-coated plates[16] or in a cold atomic sample of dilute Rydberg atoms trapped in front of a rough substrate.[12] In future, we will apply the completeness relation of continuous eigenvalues in the Gamma-ray burst, where the spectrum density factor shall appear naturally in a form of the exponential function as a high-energy cutoff.[32-33]
Appendix: The Completeness of the Continuum Operators
In this Appendix, we explain that the completeness of the continuum operators could be achieved by a transformation. We illustrate the idea by first discussing it for a discrete number operator $\hat{n}=\hat{a}^\dagger\hat{a}$, which is Hermitian and has the following eigenvector set $\{\vert n\rangle^{(1)} \}$,
$$ \vert n\rangle^{(1)} =(\hat{a}^\dagger)^n \vert 0\rangle\,. $$
However, this set does not satisfy the completeness relation,
$ \sum_n \vert n\rangle^{(1)} {}^{(1)}\langle n\vert=\hat{I}\,, $
due to the fact that $\vert n\rangle^{(1)} $ is not normalized. A simple transformation solves the problem by introducing a "spectrum density factor" $f(n)$ (here it is simply a normalized constant),
$ \vert n\rangle=f(n)\vert n\rangle^{(1)}=f(n)(\hat{a}^\dagger)^n \vert 0\rangle\,, $
with $f(n)=1/\sqrt{n!}$. $\{\vert n\rangle \}$ forms a complete set of eigenstates
$ \sum_n \vert n\rangle\langle n\vert=\hat{I}~. $
Thus, for the discrete spectra of observables, it is easy to form a complete set of eigenstates by normalizing them. However, the eigenvectors of the continuum spectra are not normalizable. Then the question arises on how to make them a complete set of eigenstates. The key is to generalize the above transformation $f(n)$. We clarify this point by a pair of conjugate operators $(\hat{x},\hat{p})$. By introducing the bosonic annihilation operator $\hat{a}$ and creation operators $ \hat{a}^\dagger$
$$ \hat{x} = \frac{1}{\sqrt{2}} (\hat{a}^\dagger+\hat{a})\,,\\ \hat{p} = \frac{{\rm i}}{\sqrt{2}} (\hat{a}^\dagger-\hat{a})\,. $$
Here, $[\hat{a}, \hat{a}^\dagger]=1$. In the $(\hat{a}, \hat{a}^\dagger)$ space, the set of eigenstates of $(\hat{x},\hat{p})$ can be expressed in details as
$ \vert {x}\rangle^{(1)} = \exp\Bigl[-\frac{1}{2}\hat{a}^\dagger\hat{a}^\dagger+\sqrt{2}x \hat{a}^\dagger\Bigr] \vert 0\rangle\,,\\ \vert {p}\rangle^{(1)} = \exp\Bigl[\frac{1}{2}\hat{a}^\dagger\hat{a}^\dagger+{\rm i}\sqrt{2}p \hat{a}^\dagger\Bigr] \vert 0\rangle\,. $
Here, $x$ and $p$ are eigenvalues of $\hat{x}$ and $\hat{p}$, respectively, satisfying,
$$\hat{x}\vert {x}\rangle^{(1)} =\frac{1}{\sqrt{2}}(\hat{a}^\dagger+\hat{a})\exp \Bigl[-\frac{1}{2}\hat{a}^\dagger\hat{a}^\dagger+\sqrt{2}x \hat{a}^\dagger\Bigr] \vert 0\rangle\nonumber\\ =x\vert {x}\rangle^{(1)}\,,\nonumber\\ \hat{p}\vert {p}\rangle^{(1)} =\frac{i}{\sqrt{2}}(\hat{a}^\dagger -\hat{a})\exp\Bigl[\frac{1}{2}\hat{a}^\dagger \hat{a}^\dagger+{\rm i}\sqrt{2}p \hat{a}^\dagger\Bigr] \vert 0\rangle\nonumber\\ =p\vert {p}\rangle^{(1)}\,.\nonumber $$
Equation (A4) is the eigenvector set of $(\hat{x},\hat{p})$. However, the present expressions of $\vert {x}\rangle^{(1)}$ and $\vert {p}\rangle^{(1)}$ obviously do not satisfy the completeness relations. Similar to Eq. (A2), we need to introduce the "normalization coefficients'' $F_1(x)$ and $F_2(p)$
$$ \vert {x}\rangle =F_1(x) \vert {x}\rangle^{(1)}\,,\\ \vert {p}\rangle =F_2(p)\vert {p}\rangle^{(1)}\,, $$
which we call in this paper the spectrum density factor. Here, the set of basis kets of $\vert {x}\rangle$ and $\vert {p}\rangle$ satisfy the completeness relation,
$ \int {\rm d}x \vert x\rangle\langle x\vert= \hat{I}\,,\quad \int {\rm d}p \vert p\rangle\langle p\vert= \hat{I}\,. $
In the following, we derive the detailed expression of $F_1(x)$. Making use of Eq. (A2), we project the vector $\vert n\rangle$ into $x$ space,
$ \langle x\vert n\rangle =\frac{1}{\sqrt{n!}}\int^{\infty}_{-\infty}\langle x\vert \hat{a}^{\dagger n}\vert x'\rangle \langle x' \vert 0\rangle {\rm d}x' \\ = \frac{1}{\sqrt{2^n n!}} \int^{\infty}_{-\infty} {\rm d}x' \Bigl( x-\frac{{\rm d}}{{\rm d}x}\Bigr)^n \delta(x-x') \langle x' \vert 0\rangle \\ = \frac{1}{\sqrt{\sqrt{\pi}2^n n!}}\Bigl( x-\frac{{\rm d}}{{\rm d}x}\Bigr)^n {\rm e}^{-x^2/2}\,. $
Making use of the Hermite polynomial,
$$ H_n(x)={\rm e}^{x^2/2}\Bigl( x-\frac{{\rm d}}{{\rm d}x}\Bigr)^n {\rm e}^{-x^2/2}\,, $$
we achieve,
$ \langle x\vert n\rangle =\frac{1}{\sqrt{\sqrt{\pi}2^n n!}}{\rm e}^{-x^2/2} H_n(x)\,. $
Combined with the complete relation Eq. (A1), the Fock representation of the position operator is obtained,[2]
$$ \vert x\rangle =\pi^{-1/4} \exp\Bigl[-\frac{x^2}{2}-\frac{1}{2}\hat{a}^\dagger \hat{a}^\dagger+\sqrt{2}x \hat{a}^\dagger\Bigr] \vert 0\,. $$
Comparing to Eq. (A5), the spectrum density factor $F_i(x), i=1, 2$ is,
$$ F_1(x)=\pi^{-1/4} \exp\Bigl[-\frac{x^2}{2}\Bigr]\,. $$
Similarly,
$$ F_2(p)=\pi^{-1/4} \exp\Bigl[-\frac{p^2}{2}\Bigr]\,. $$
From the above example, we learn that an observable has a set of eigenstates, which is orthogonal but not necessarily complete. An incomplete eigenvectors of observable can be transformed into a complete set by introducing a spectrum density factor, which is unique.
In the main text, we will use the same technique for the photons in the free space, where the eigenvector set of its momentum and energy is, $\vert { k}\rangle \sim {\rm e}^{-{\rm i}{ k}\cdot { x}} $, and is not normalizable due to the nonconservation of the photon numbers. This eigenstate will sometime lead to divergence for the physical quantity based on it. The solution is to introduce a similar spectrum density factor depending on $k$ to eradicate the divergence in the Casimir effect. Furthermore, we apply it to the Lamb shift.