Quantization of the electromagnetic field

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https://en.wikipedia.org/wiki/Quantization_of_the_electromagnetic_field

The quantization of the electromagnetic field means that an electromagnetic field consists of discrete energy parcels called photons. Photons are massless particles of definite energy, definite momentum, and definite spin.

To explain the photoelectric effect, Albert Einstein assumed heuristically in 1905 that an electromagnetic field consists of particles of energy of amount hν, where h is Planck's constant and ν is the wave frequency. In 1927 Paul A. M. Dirac was able to weave the photon concept into the fabric of the new quantum mechanics and to describe the interaction of photons with matter. He applied a technique which is now generally called second quantization, although this term is somewhat of a misnomer for electromagnetic fields, because they are solutions of the classical Maxwell equations. In Dirac's theory the fields are quantized for the first time and it is also the first time that Planck's constant enters the expressions. In his original work, Dirac took the phases of the different electromagnetic modes (Fourier components of the field) and the mode energies as dynamic variables to quantize (i.e., he reinterpreted them as operators and postulated commutation relations between them). At present it is more common to quantize the Fourier components of the vector potential. This is what is done below.

A quantum mechanical photon state ${\displaystyle |\mathbf{k},\mu ⟩}$ belonging to mode ${\displaystyle (\mathbf{k},\mu )}$ is introduced below, and it is shown that it has the following properties:

${\displaystyle \begin{array}{rl}{m}_{\text{photon}}& =0\\ H|\mathbf{k},\mu ⟩& =h\nu |\mathbf{k},\mu ⟩& & \text{with}\nu =c|\mathbf{k}|\\ P_{\text{EM}}|\mathbf{k},\mu ⟩& =\hslash \mathbf{k}|\mathbf{k},\mu ⟩\\ S_z|\mathbf{k},\mu ⟩& =\mu |\mathbf{k},\mu ⟩& & \mu =\pm 1.\end{array}}$

These equations say respectively: a photon has zero rest mass; the photon energy is hν = hc|k| (k is the wave vector, c is speed of light); its electromagnetic momentum is ℏk [ℏ=h/(2π)]; the polarization μ = ±1 is the eigenvalue of the z-component of the photon spin.

Second quantization

Second quantization starts with an expansion of a scalar or vector field (or wave functions) in a basis consisting of a complete set of functions. These expansion functions depend on the coordinates of a single particle. The coefficients multiplying the basis functions are interpreted as operators and (anti)commutation relations between these new operators are imposed, commutation relations for bosons and anticommutation relations for fermions (nothing happens to the basis functions themselves). By doing this, the expanded field is converted into a fermion or boson operator field. The expansion coefficients have been promoted from ordinary numbers to operators, creation and annihilation operators. A creation operator creates a particle in the corresponding basis function and an annihilation operator annihilates a particle in this function.

In the case of EM fields the required expansion of the field is the Fourier expansion.

Electromagnetic field and vector potential

As the term suggests, an EM field consists of two vector fields, an electric field ${\displaystyle \mathbf{E}(\mathbf{r},t)}$ and a magnetic field ${\displaystyle \mathbf{B}(\mathbf{r},t)}$. Both are time-dependent vector fields that in vacuum depend on a third vector field ${\displaystyle \mathbf{A}(\mathbf{r},t)}$ (the vector potential), as well as a scalar field ${\displaystyle \varphi (\mathbf{r},t)}$

${\displaystyle \begin{array}{rl}\mathbf{B}(\mathbf{r},t)& =\mathbf{\nabla }\times \mathbf{A}(\mathbf{r},t)\\ \mathbf{E}(\mathbf{r},t)& =-\mathbf{\nabla }\varphi (\mathbf{r},t)-\frac{\mathrm{\partial }\mathbf{A}(\mathbf{r},t)}{\mathrm{\partial }t},\end{array}}$

where ∇ × A is the curl of A.

Choosing the Coulomb gauge, for which ∇⋅A = 0, makes A into a transverse field. The Fourier expansion of the vector potential enclosed in a finite cubic box of volume V = L³ is then

${\displaystyle \mathbf{A}(\mathbf{r},t)=\sum _{\mathbf{k}}\sum _{\mu =\pm 1}\left({\mathbf{e}}^{(\mu )}(\mathbf{k})a_{\mathbf{k}}^{(\mu )}(t)e^{i\mathbf{k}\cdot \mathbf{r}}+{\overline{\mathbf{e}}}^{(\mu )}(\mathbf{k}){\overline{a}}_{\mathbf{k}}^{(\mu )}(t)e^{-i\mathbf{k}\cdot \mathbf{r}}\right),}$

where ${\displaystyle \overline{a}}$ denotes the complex conjugate of ${\displaystyle a}$. The wave vector k gives the propagation direction of the corresponding Fourier component (a polarized monochromatic wave) of A(r,t); the length of the wave vector is

${\displaystyle |\mathbf{k}|=\frac{2\pi \nu }{c}=\frac{\omega }{c},}$

with ν the frequency of the mode. In this summation k runs over all integers, both positive and negative. (The component of Fourier basis ${\displaystyle e^{-i\mathbf{k}\cdot \mathbf{r}}}$ is complex conjugate of component of ${\displaystyle e^{i\mathbf{k}\cdot \mathbf{r}}}$ as ${\displaystyle \mathbf{A}(\mathbf{r},t)}$ is real.) The components of the vector k have discrete values (a consequence of the boundary condition that A has the same value on opposite walls of the box):

${\displaystyle k_x=\frac{2\pi n_x}{L},k_y=\frac{2\pi n_y}{L},k_z=\frac{2\pi n_z}{L},n_x,n_y,n_z=0,\pm 1,\pm 2,\dots .}$

Two e^(μ) ("polarization vectors") are conventional unit vectors for left and right hand circular polarized (LCP and RCP) EM waves (See Jones calculus or Jones vector, Jones calculus) and perpendicular to k. They are related to the orthonormal Cartesian vectors e_x and e_y through a unitary transformation,

${\displaystyle {\mathbf{e}}^{(\pm 1)}\equiv \frac{\mp 1}{\sqrt{2}}({\mathbf{e}}_x\pm i{\mathbf{e}}_y)\text{with}{\mathbf{e}}_x\cdot \mathbf{k}={\mathbf{e}}_y\cdot \mathbf{k}=0.}$

The k-th Fourier component of A is a vector perpendicular to k and hence is a linear combination of e⁽¹⁾ and e⁽⁻¹⁾. The superscript μ indicates a component along e^(μ).

Clearly, the (discrete infinite) set of Fourier coefficients ${\displaystyle a_{\mathbf{k}}^{(\mu )}(t)}$ and ${\displaystyle {\overline{a}}_{\mathbf{k}}^{(\mu )}(t)}$ are variables defining the vector potential. In the following they will be promoted to operators.

By using field equations of ${\displaystyle \mathbf{B}}$ and ${\displaystyle \mathbf{E}}$ in terms of ${\displaystyle \mathbf{A}}$ above, electric and magnetic fields are

${\displaystyle \begin{array}{rl}\mathbf{E}(\mathbf{r},t)& =i\sum _{\mathbf{k}}\sum _{\mu =\pm 1}\omega \left({\mathbf{e}}^{(\mu )}(\mathbf{k})a_{\mathbf{k}}^{(\mu )}(t)e^{i\mathbf{k}\cdot \mathbf{r}}-{\overline{\mathbf{e}}}^{(\mu )}(\mathbf{k}){\overline{a}}_{\mathbf{k}}^{(\mu )}(t)e^{-i\mathbf{k}\cdot \mathbf{r}}\right)\\ \mathbf{B}(\mathbf{r},t)& =i\sum _{\mathbf{k}}\sum _{\mu =\pm 1}\left\{\left(\mathbf{k}\times {\mathbf{e}}^{(\mu )}(\mathbf{k})\right)a_{\mathbf{k}}^{(\mu )}(t)e^{i\mathbf{k}\cdot \mathbf{r}}-\left(\mathbf{k}\times {\overline{\mathbf{e}}}^{(\mu )}(\mathbf{k})\right){\overline{a}}_{\mathbf{k}}^{(\mu )}(t)e^{-i\mathbf{k}\cdot \mathbf{r}}\right\}\end{array}}$

By using identity ${\displaystyle \mathrm{\nabla }\times e^{A\cdot r}=A\times e^{A\cdot r}}$ (${\displaystyle A}$ and ${\displaystyle r}$ are vectors) and ${\displaystyle a_{\mathbf{k}}^{(\mu )}(t)=a_{\mathbf{k}}^{(\mu )}{e}^{-iwt}}$ as each mode has single frequency dependence.

Quantization of EM field

The best known example of quantization is the replacement of the time-dependent linear momentum of a particle by the rule

${\displaystyle \mathbf{p}(t)\to -i\hslash \mathbf{\nabla }.}$

Note that Planck's constant is introduced here and that the time-dependence of the classical expression is not taken over in the quantum mechanical operator (this is true in the so-called Schrödinger picture).

For the EM field we do something similar. The quantity ${\displaystyle {ϵ}_0}$ is the electric constant, which appears here because of the use of electromagnetic SI units. The quantization rules are:

${\displaystyle \begin{array}{rl}{a}_{\mathbf{k}}^{(\mu )}(t)& \to \sqrt{\frac{\hslash }{2\omega V{ϵ}_0}}{a}^{(\mu )}(\mathbf{k})\\ {\overline{a}}_{\mathbf{k}}^{(\mu )}(t)& \to \sqrt{\frac{\hslash }{2\omega V{ϵ}_0}}{a^{†}}^{(\mu )}(\mathbf{k})\end{array}}$

subject to the boson commutation relations

${\displaystyle \begin{array}{rl}\left[a^{(\mu )}(\mathbf{k}),a^{({\mu }^{\prime })}({\mathbf{k}}^{\prime })\right]& =0\\ \left[{a^{†}}^{(\mu )}(\mathbf{k}),{a^{†}}^{({\mu }^{\prime })}({\mathbf{k}}^{\prime })\right]& =0\\ \left[a^{(\mu )}(\mathbf{k}),{a^{†}}^{({\mu }^{\prime })}({\mathbf{k}}^{\prime })\right]& ={\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}{\delta }_{\mu ,{\mu }^{\prime }}\end{array}}$

The square brackets indicate a commutator, defined by ${\displaystyle [A,B]\equiv AB-BA}$ for any two quantum mechanical operators A and B. The introduction of Planck's constant is essential in the transition from a classical to a quantum theory. The factor

${\displaystyle \sqrt{\frac{1}{2\omega V{ϵ}_0}}}$

is introduced to give the Hamiltonian (energy operator) a simple form, see below.

The quantized fields (operator fields) are the following

${\displaystyle \begin{array}{rl}\mathbf{A}(\mathbf{r})& =\sum _{\mathbf{k},\mu }\sqrt{\frac{\hslash }{2\omega V{ϵ}_0}}\left\{{\mathbf{e}}^{(\mu )}{a}^{(\mu )}(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{r}}+{\overline{\mathbf{e}}}^{(\mu )}{a^{†}}^{(\mu )}(\mathbf{k})e^{-i\mathbf{k}\cdot \mathbf{r}}\right\}\\ \mathbf{E}(\mathbf{r})& =i\sum _{\mathbf{k},\mu }\sqrt{\frac{\hslash \omega }{2V{ϵ}_0}}\left\{{\mathbf{e}}^{(\mu )}{a}^{(\mu )}(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{r}}-{\overline{\mathbf{e}}}^{(\mu )}{a^{†}}^{(\mu )}(\mathbf{k})e^{-i\mathbf{k}\cdot \mathbf{r}}\right\}\\ \mathbf{B}(\mathbf{r})& =i\sum _{\mathbf{k},\mu }\sqrt{\frac{\hslash }{2\omega V{ϵ}_0}}\left\{\left(\mathbf{k}\times {\mathbf{e}}^{(\mu )}\right)a^{(\mu )}(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{r}}-\left(\mathbf{k}\times {\overline{\mathbf{e}}}^{(\mu )}\right){a^{†}}^{(\mu )}(\mathbf{k})e^{-i\mathbf{k}\cdot \mathbf{r}}\right\}\end{array}}$

where ω = c |k| = ck.

Hamiltonian of the field

The classical Hamiltonian has the form

${\displaystyle H=\frac{1}{2}{ϵ}_0{\iiint }_V\left({\left|E(\mathbf{r},t)\right|}^2+c^2{\left|B(\mathbf{r},t)\right|}^2\right){\text{d}}^3\mathbf{r}=V{ϵ}_0\sum _{\mathbf{k}}\sum _{\mu =\pm 1}{\omega }^2\left({\overline{a}}_{\mathbf{k}}^{(\mu )}(t)a_{\mathbf{k}}^{(\mu )}(t)+a_{\mathbf{k}}^{(\mu )}(t){\overline{a}}_{\mathbf{k}}^{(\mu )}(t)\right).}$

The right-hand-side is easily obtained by first using

${\displaystyle {\int }_V{e}^{ik\cdot r}{e}^{-i{k}^{\prime }\cdot r}dr=V{\delta }_{k,k^{\prime }}}$

(can be derived from Euler equation and trigonometric orthogonality) where k is wavenumber for wave confined within the box of V = L × L × L as described above and second, using ω = kc.

Substitution of the field operators into the classical Hamiltonian gives the Hamilton operator of the EM field,

${\displaystyle H=\frac{1}{2}\sum _{\mathbf{k},\mu =\pm 1}\hslash \omega \left({a^{†}}^{(\mu )}(\mathbf{k})a^{(\mu )}(\mathbf{k})+a^{(\mu )}(\mathbf{k}){a^{†}}^{(\mu )}(\mathbf{k})\right)=\sum _{\mathbf{k},\mu }\hslash \omega \left({a^{†}}^{(\mu )}(\mathbf{k})a^{(\mu )}(\mathbf{k})+\frac{1}{2}\right)}$

The second equality follows by use of the third of the boson commutation relations from above with k′ = k and μ′ = μ. Note again that ℏω = hν = ℏc|k| and remember that ω depends on k, even though it is not explicit in the notation. The notation ω(k) could have been introduced, but is not common as it clutters the equations.

Digression: harmonic oscillator

The second quantized treatment of the one-dimensional quantum harmonic oscillator is a well-known topic in quantum mechanical courses. We digress and say a few words about it. The harmonic oscillator Hamiltonian has the form

${\displaystyle H=\hslash \omega \left(a^{†}a+\frac{1}{2}\right)}$

where ω ≡ 2πν is the fundamental frequency of the oscillator. The ground state of the oscillator is designated by ${\displaystyle |0⟩}$; and is referred to as the "vacuum state". It can be shown that ${\displaystyle a^{†}}$ is an excitation operator, it excites from an n fold excited state to an n + 1 fold excited state:

${\displaystyle a^{†}|n⟩=|n+1⟩\sqrt{n+1}.}$

In particular: ${\displaystyle a^{†}|0⟩=|1⟩}$ and ${\displaystyle (a^{†}{)}^n|0⟩\propto |n⟩.}$

Since harmonic oscillator energies are equidistant, the n-fold excited state ${\displaystyle |n⟩}$; can be looked upon as a single state containing n particles (sometimes called vibrons) all of energy hν. These particles are bosons. For obvious reason the excitation operator ${\displaystyle a^{†}}$ is called a creation operator.

From the commutation relation follows that the Hermitian adjoint ${\displaystyle a}$ de-excites: ${\displaystyle a|n⟩=|n-1⟩\sqrt{n}}$ in particular ${\displaystyle a|0⟩\propto 0,}$ so that ${\displaystyle a|0⟩=0.}$ For obvious reason the de-excitation operator ${\displaystyle a}$ is called an annihilation operator.

By mathematical induction the following "differentiation rule", that will be needed later, is easily proved,

${\displaystyle \left[a,(a^{†}{)}^n\right]=n(a^{†}{)}^{n-1}\text{with}{\left(a^{†}\right)}^0=1.}$

Suppose now we have a number of non-interacting (independent) one-dimensional harmonic oscillators, each with its own fundamental frequency ω_i . Because the oscillators are independent, the Hamiltonian is a simple sum:

${\displaystyle H=\sum _i\hslash {\omega }_i\left(a^{†}(i)a(i)+\frac{1}{2}\right).}$

By substituting ${\displaystyle (\mathbf{k},\mu )}$ for ${\displaystyle i}$ we see that the Hamiltonian of the EM field can be considered a Hamiltonian of independent oscillators of energy ω = |k|c oscillating along direction e^(μ) with μ = ±1.

Photon number states (Fock states)

The quantized EM field has a vacuum (no photons) state ${\displaystyle |0⟩}$. The application to it of, say,

${\displaystyle {\left({a^{†}}^{(\mu )}(\mathbf{k})\right)}^m{\left({a^{†}}^{({\mu }^{\prime })}({\mathbf{k}}^{\prime })\right)}^n|0⟩\propto |(\mathbf{k},\mu {)}^m;({\mathbf{k}}^{\prime },{\mu }^{\prime }{)}^n⟩,}$

gives a quantum state of m photons in mode (k, μ) and n photons in mode (k′, μ′). The proportionality symbol is used because the state on the left-hand is not normalized to unity, whereas the state on the right-hand may be normalized.

The operator

${\displaystyle N^{(\mu )}(\mathbf{k})\equiv {a^{†}}^{(\mu )}(\mathbf{k})a^{(\mu )}(\mathbf{k})}$

is the number operator. When acting on a quantum mechanical photon number state, it returns the number of photons in mode (k, μ). This also holds when the number of photons in this mode is zero, then the number operator returns zero. To show the action of the number operator on a one-photon ket, we consider

${\displaystyle \begin{array}{rl}{N}^{(\mu )}(\mathbf{k})|{\mathbf{k}}^{\prime },{\mu }^{\prime }⟩& ={a^{†}}^{(\mu )}(\mathbf{k})a^{(\mu )}(\mathbf{k}){a^{†}}^{({\mu }^{\prime })}({\mathbf{k}}^{\prime })|0⟩\\ & ={a^{†}}^{(\mu )}(\mathbf{k})\left({\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}{\delta }_{\mu ,{\mu }^{\prime }}+{a^{†}}^{({\mu }^{\prime })}({\mathbf{k}}^{\prime })a^{(\mu )}(\mathbf{k})\right)|0⟩\\ & ={\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}{\delta }_{\mu ,{\mu }^{\prime }}|\mathbf{k},\mu ⟩,\end{array}}$

i.e., a number operator of mode (k, μ) returns zero if the mode is unoccupied and returns unity if the mode is singly occupied. To consider the action of the number operator of mode (k, μ) on a n-photon ket of the same mode, we drop the indices k and μ and consider

${\displaystyle N(a^{†}{)}^n|0⟩=a^{†}\left([a,(a^{†}{)}^n]+(a^{†}{)}^{n}a\right)|0⟩=a^{†}[a,(a^{†}{)}^n]|0⟩.}$

Use the "differentiation rule" introduced earlier and it follows that

${\displaystyle N(a^{†}{)}^n|0⟩=n(a^{†}{)}^n|0⟩.}$

A photon number state (or a Fock state) is an eigenstate of the number operator. This is why the formalism described here is often referred to as the occupation number representation.

Photon energy

Earlier the Hamiltonian,

${\displaystyle H=\sum _{\mathbf{k},\mu }\hslash \omega \left({a^{†}}^{(\mu )}(\mathbf{k})a^{(\mu )}(\mathbf{k})+\frac{1}{2}\right)}$

was introduced. The zero of energy can be shifted, which leads to an expression in terms of the number operator,

${\displaystyle H=\sum _{\mathbf{k},\mu }\hslash \omega N^{(\mu )}(\mathbf{k})}$

The effect of H on a single-photon state is

${\displaystyle H|\mathbf{k},\mu ⟩\equiv H\left({a^{†}}^{(\mu )}(\mathbf{k})|0⟩\right)=\sum _{{\mathbf{k}}^{\prime },{\mu }^{\prime }}\hslash {\omega }^{\prime }{N}^{({\mu }^{\prime })}({\mathbf{k}}^{\prime }){a^{†}}^{(\mu )}(\mathbf{k})|0⟩=\hslash \omega \left({a^{†}}^{(\mu )}(\mathbf{k})|0⟩\right)=\hslash \omega |\mathbf{k},\mu ⟩.}$

Apparently, the single-photon state is an eigenstate of H and ℏω = hν is the corresponding energy. In the very same way

${\displaystyle H|(\mathbf{k},\mu {)}^m;({\mathbf{k}}^{\prime },{\mu }^{\prime }{)}^n⟩=\left[m(\hslash \omega )+n(\hslash {\omega }^{\prime })\right]|(\mathbf{k},\mu {)}^m;({\mathbf{k}}^{\prime },{\mu }^{\prime }{)}^n⟩,\text{with}\omega =c|\mathbf{k}|\text{and}{\omega }^{\prime }=c|{\mathbf{k}}^{\prime }|.}$

Example photon density

The electromagnetic energy density created by a 100 kW radio transmitting station is computed in the article on the electromagnetic wave (where?) ; the energy density estimate at 5 km from the station was 2.1 × 10⁻¹⁰ J/m³. Is quantum mechanics needed to describe the station's broadcast?

The classical approximation to EM radiation is good when the number of photons is much larger than unity in the volume ${\displaystyle \frac{{\lambda }^3}{(2\pi {)}^3},}$ where λ is the length of the radio waves. In that case quantum fluctuations are negligible and cannot be heard.

Suppose the radio station broadcasts at ν = 100 MHz, then it is sending out photons with an energy content of νh = 1 × 10⁸ × 6.6 × 10⁻³⁴ = 6.6 × 10⁻²⁶ J, where h is Planck's constant. The wavelength of the station is λ = c/ν = 3 m, so that λ/(2π) = 48 cm and the volume is 0.109 m³. The energy content of this volume element is 2.1 × 10⁻¹⁰ × 0.109 = 2.3 × 10⁻¹¹ J, which amounts to 3.4 × 10¹⁴ photons per ${\displaystyle \frac{{\lambda }^3}{(2\pi {)}^3}.}$ Obviously, 3.4 × 10¹⁴ > 1 and hence quantum effects do not play a role; the waves emitted by this station are well-described by the classical limit and quantum mechanics is not needed.

Photon momentum

Introducing the Fourier expansion of the electromagnetic field into the classical form

${\displaystyle {\mathbf{P}}_{\text{EM}}={ϵ}_0{\iiint }_V\mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t){\text{d}}^3\mathbf{r},}$

yields

${\displaystyle {\mathbf{P}}_{\text{EM}}=V{ϵ}_0\sum _{\mathbf{k}}\sum _{\mu =1,-1}\omega \mathbf{k}\left(a_{\mathbf{k}}^{(\mu )}(t){\overline{a}}_{\mathbf{k}}^{(\mu )}(t)+{\overline{a}}_{\mathbf{k}}^{(\mu )}(t)a_{\mathbf{k}}^{(\mu )}(t)\right).}$

Quantization gives

${\displaystyle {\mathbf{P}}_{\text{EM}}=\sum _{\mathbf{k},\mu }\hslash \mathbf{k}\left({a^{†}}^{(\mu )}(\mathbf{k})a^{(\mu )}(\mathbf{k})+\frac{1}{2}\right)=\sum _{\mathbf{k},\mu }\hslash \mathbf{k}{N}^{(\mu )}(\mathbf{k}).}$

The term 1/2 could be dropped, because when one sums over the allowed k, k cancels with −k. The effect of P_EM on a single-photon state is

${\displaystyle {\mathbf{P}}_{\text{EM}}|\mathbf{k},\mu ⟩={\mathbf{P}}_{\text{EM}}\left({a^{†}}^{(\mu )}(\mathbf{k})|0⟩\right)=\hslash \mathbf{k}\left({a^{†}}^{(\mu )}(\mathbf{k})|0⟩\right)=\hslash \mathbf{k}|\mathbf{k},\mu ⟩.}$

Apparently, the single-photon state is an eigenstate of the momentum operator, and ℏk is the eigenvalue (the momentum of a single photon).

Photon mass

The photon having non-zero linear momentum, one could imagine that it has a non-vanishing rest mass m₀, which is its mass at zero speed. However, we will now show that this is not the case: m₀ = 0.

Since the photon propagates with the speed of light, special relativity is called for. The relativistic expressions for energy and momentum squared are,

${\displaystyle E^2=\frac{m_0^2{c}^4}{1-v^2/c^2},p^2=\frac{m_0^2{v}^2}{1-v^2/c^2}.}$

From p²/E²,

${\displaystyle \frac{v^2}{c^2}=\frac{c^2{p}^2}{E^2}⟹E^2=\frac{m_0^2{c}^4}{1-c^2{p}^2/E^2}⟹m_0^2{c}^4=E^2-c^2{p}^2.}$

Use

${\displaystyle E^2={\hslash }^2{\omega }^2\text{and}{p}^2={\hslash }^2{k}^2=\frac{{\hslash }^2{\omega }^2}{c^2}}$

and it follows that

${\displaystyle m_0^2{c}^4=E^2-c^2{p}^2={\hslash }^2{\omega }^2-c^2\frac{{\hslash }^2{\omega }^2}{c^2}=0,}$

so that m₀ = 0.

Photon spin

The photon can be assigned a triplet spin with spin quantum number S = 1. This is similar to, say, the nuclear spin of the ¹⁴N isotope, but with the important difference that the state with M_S = 0 is zero, only the states with M_S = ±1 are non-zero.

Define spin operators:

${\displaystyle S_z\equiv -i\hslash \left({\mathbf{e}}_x\otimes {\mathbf{e}}_y-{\mathbf{e}}_y\otimes {\mathbf{e}}_x\right)\text{and cyclically}x\to y\to z\to x.}$

The two operators ${\displaystyle \otimes }$ between the two orthogonal unit vectors are dyadic products. The unit vectors are perpendicular to the propagation direction k (the direction of the z axis, which is the spin quantization axis).

The spin operators satisfy the usual angular momentum commutation relations

${\displaystyle [S_x,S_y]=i\hslash S_z\text{and cyclically}x\to y\to z\to x.}$

Indeed, use the dyadic product property

${\displaystyle \left({\mathbf{e}}_y\otimes {\mathbf{e}}_z\right)\left({\mathbf{e}}_z\otimes {\mathbf{e}}_x\right)=\left({\mathbf{e}}_y\otimes {\mathbf{e}}_x\right)\left({\mathbf{e}}_z\cdot {\mathbf{e}}_z\right)={\mathbf{e}}_y\otimes {\mathbf{e}}_x}$

because e_z is of unit length. In this manner,

${\displaystyle \begin{array}{rl}\left[S_x,S_y\right]& =-{\hslash }^2\left({\mathbf{e}}_y\otimes {\mathbf{e}}_z-{\mathbf{e}}_z\otimes {\mathbf{e}}_y\right)\left({\mathbf{e}}_z\otimes {\mathbf{e}}_x-{\mathbf{e}}_x\otimes {\mathbf{e}}_z\right)+{\hslash }^2\left({\mathbf{e}}_z\otimes {\mathbf{e}}_x-{\mathbf{e}}_x\otimes {\mathbf{e}}_z\right)\left({\mathbf{e}}_y\otimes {\mathbf{e}}_z-{\mathbf{e}}_z\otimes {\mathbf{e}}_y\right)\\ & ={\hslash }^2\left[-\left({\mathbf{e}}_y\otimes {\mathbf{e}}_z-{\mathbf{e}}_z\otimes {\mathbf{e}}_y\right)\left({\mathbf{e}}_z\otimes {\mathbf{e}}_x-{\mathbf{e}}_x\otimes {\mathbf{e}}_z\right)+\left({\mathbf{e}}_z\otimes {\mathbf{e}}_x-{\mathbf{e}}_x\otimes {\mathbf{e}}_z\right)\left({\mathbf{e}}_y\otimes {\mathbf{e}}_z-{\mathbf{e}}_z\otimes {\mathbf{e}}_y\right)\right]\\ & =i\hslash \left[-i\hslash \left({\mathbf{e}}_x\otimes {\mathbf{e}}_y-{\mathbf{e}}_y\otimes {\mathbf{e}}_x\right)\right]\\ & =i\hslash S_z\end{array}}$

By inspection it follows that

${\displaystyle -i\hslash \left({\mathbf{e}}_x\otimes {\mathbf{e}}_y-{\mathbf{e}}_y\otimes {\mathbf{e}}_x\right)\cdot {\mathbf{e}}^{(\mu )}=\mu \hslash {\mathbf{e}}^{(\mu )},\mu =\pm 1,}$

and therefore μ labels the photon spin,

${\displaystyle S_z|\mathbf{k},\mu ⟩=\mu \hslash |\mathbf{k},\mu ⟩,\mu =\pm 1.}$

Because the vector potential A is a transverse field, the photon has no forward (μ = 0) spin component.

Thiomersal and vaccines

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Thiomersal_and_vaccines

Thiomersal (or thimerosal) is a mercury compound which is used as a preservative in some vaccines. Anti-vaccination activists promoting the incorrect claim that vaccination causes autism have asserted that the mercury in thiomersal is the cause. There is no scientific evidence to support this claim. The idea that thiomersal in vaccines might have detrimental effects originated with anti-vaccination activists and was sustained by them and especially through the action of plaintiffs' lawyers.

The potential impact of thiomersal on autism has been investigated extensively. Multiple lines of scientific evidence have shown that thiomersal does not cause autism. For example, the clinical symptoms of mercury poisoning differ significantly from those of autism. In addition, multiple population studies have found no association between thiomersal and autism, and rates of autism have continued to increase despite removal of thiomersal from vaccines. Thus, major scientific and medical bodies such as the Institute of Medicine and World Health Organization (WHO) as well as governmental agencies such as the Food and Drug Administration (FDA) and the Centers for Disease Control and Prevention (CDC) reject any role for thiomersal in autism or other neurodevelopmental disorders. In spite of the consensus of the scientific community, some parents and advocacy groups continue to contend that thiomersal is linked to autism and the claim is still stated as if it were fact in anti-vaccination propaganda, notably that of Robert F. Kennedy Jr., through his group Children's Health Defense. Thiomersal is no longer used in most children's vaccines in the United States, with the exception of some types of flu shots. While exposure to mercury may result in damage to brain, kidneys, and developing fetus, the scientific consensus is that thiomersal has no such effects.

This controversy has caused harm due to parents attempting to treat their autistic children with unproven and possibly dangerous treatments, discouraging parents from vaccinating their children due to fears about thiomersal toxicity and diverting resources away from research into more promising areas for the cause of autism. Thousands of lawsuits have been filed in the U.S. to seek damages from alleged toxicity from vaccines, including those purportedly caused by thiomersal. US courts have ruled against multiple representative test cases involving thiomersal. A 2011 journal article described the vaccine-autism connection as "perhaps, the most damaging medical hoax of the last 100 years".

History

Thiomersal (also spelled thimerosal, especially in the United States) is an organomercury compound used as a preservative in vaccines to prevent bacterial and fungal contamination. Following a mandated review of mercury-containing food and drugs in 1999, the Centers for Disease Control and Prevention (CDC) and the American Academy of Pediatrics (AAP) determined that under the existing vaccination schedule "some children could be exposed to a cumulative level of mercury over the first 6 months of life that exceeds one of the federal guidelines on methyl mercury." They asked vaccine makers to remove thiomersal from vaccines as quickly as possible as a precautionary measure, and it was rapidly phased out of most US and EU vaccines, but is still used in multi-dose vials of flu vaccines in the U.S. No vaccines in the European Union currently contain thiomersal as a preservative. In the context of perceived increased autism rates and increased number of vaccines in the childhood vaccination schedule, some parents believed the action to remove thiomersal was an indication that the preservative caused autism.

It was introduced as a preservative in the 1930s to prevent the growth of infectious organisms such as bacteria and fungi, and has been in use in vaccines and other products such as immunoglobulin preparations and ophthalmic and nasal solutions. Vaccine manufacturers have used preservatives to prevent microbial growth during the manufacturing process or when packaged as "multi-dose" products to allow for multiple punctures of the same vial to dispense multiple vaccinations with less fear of contamination. After the FDA Modernization Act of 1997 mandated a review and risk assessment of all mercury-containing food and drugs, vaccine manufacturers responded to FDA requests made in December 1998 and April 1999 to provide detailed information about the thiomersal content of their preparations.

A review of the data showed that while the vaccine schedule for infants did not exceed FDA, Agency for Toxic Substances and Disease Registry (ATSDR), or WHO guidelines on mercury exposure, it could have exceeded Environmental Protection Agency (EPA) standards for the first six months of life, depending on the vaccine formulation and the weight of the infant. The review also highlighted difficulty interpreting toxicity of the ethylmercury in thiomersal because guidelines for mercury toxicity were based primarily on studies of methylmercury, a different mercury compound with different toxicologic properties. Multiple meetings were scheduled among various government officials and scientists from multiple agencies to discuss the appropriate response to this evidence. There was a wide range of opinions on the urgency and significance of the safety of thiomersal, with some toxicologists suggesting there was no clear evidence that thiomersal was harmful and other participants like Neal Halsey, director of the Institute of Vaccine Safety at Johns Hopkins School of Public Health, strongly advocating removal of thiomersal from vaccines due to possible safety risks. In the process of forming the response to this information, the participants attempted to strike a balance between acknowledging possible harm from thiomersal and the risks involved if childhood vaccinations were delayed or stopped.

Upon conclusion of their review, the FDA, in conjunction with the other members of the US Public Health Service (USPHS), the National Institutes of Health (NIH), CDC and Health Resources and Services Administration (HRSA), in a joint statement with the AAP in July 1999 concluded that there was "no evidence of harm caused by doses of thimerosal found in vaccines, except for local hypersensitivity reactions."

Despite the lack of convincing evidence of toxicity of thiomersal when used as a vaccine preservative, the USPHS and AAP determined that thiomersal should be removed from vaccines as a purely precautionary measure. This action was based on the precautionary principle, which assumes that there is no harm in exercising caution even if it later turns out to be unnecessary. The CDC and AAP reasoned that despite the lack of evidence of significant harm in the use of thiomersal in vaccines, the removal of this preservative would increase the public confidence in the safety of vaccines. Although thiomersal was largely removed from routine infant vaccines by summer 2001 in the U.S., some vaccines continue to contain non-trace amounts of thiomersal, mainly in multi-dose vaccines targeted against influenza, meningococcal disease and tetanus.

In 2004 Quackwatch posted an article saying that chelation therapy has been falsely promoted as effective against autism, and that practitioners falsified diagnoses of metal poisoning to "trick" parents into having their children undergo the process. As of 2008, between 2–8% of children with autism had undergone the therapy.

Rationale for concern

Reports of autism cases per 1,000 children grew dramatically in the U.S. from 1996 to 2007. It is unknown how much, if any, growth came from changes in autism's prevalence.

Although intended to increase public confidence in vaccinations, the decision to remove thiomersal instead led to some parents suspecting thiomersal as a cause of autism. This concern over a vaccine-autism link grew from a confluence of several underlying factors. First, methylmercury had for decades been the subject of widespread environmental and media concern after two highly publicized episodes of poisonings in the 1950s and 1960s in Minamata Bay, Japan from industrial waste and in the 1970s in Iraq from fungicide contamination of wheat. These incidents led to new research on methylmercury safety and culminated in the publication of an array of confusing recommendations by public health agencies in the 1990s warning against methylmercury exposure in adults and pregnant women, which ensured a continued high public awareness of mercury toxicity. Second, the vaccine schedule for infants expanded in the 1990s to include more vaccines, some of which, including the Hib vaccine, DTaP vaccine and hepatitis B vaccine, could have contained thiomersal. Third, the number of diagnoses of autism grew in the 1990s, leading parents of these children to search for an explanation for the apparent rise in diagnoses, including considering possible environmental factors. The dramatic increase in reported cases of autism during the 1990s and early 2000s is largely attributable to changes in diagnostic practices, referral patterns, availability of services, age at diagnosis, and public awareness, and it is unknown whether autism's true prevalence increased during the period. Nevertheless, some parents believed that there was a growing "autism epidemic" and connected these three factors to conclude that the increase in number of vaccines, and specifically the mercury in thiomersal in those vaccines, was causing a dramatic increase in the incidence of autism.

Advocates of a thiomersal-autism link also relied on indirect evidence from the scientific literature, including analogy with neurotoxic effects of other mercury compounds, the reported epidemiologic association between autism and vaccine use, and extrapolation from in vitro experiments and animal studies. Studies conducted by Mark Geier and his son David Geier have been the most frequently cited research by parents advocating a link between thiomersal and autism. This research by Geier has received considerable criticism for methodological problems in his research, including not presenting methods and statistical analyses to others for verification, improperly analyzing data taken from Vaccine Adverse Event Reporting System, as well as either mislabelling or confusing fundamental statistical terms in his papers, leading to results that were "uninterpretable".

Publicity of concern

Several months after the recommendation to have thiomersal removed from vaccines was published, a speculative article was published in Medical Hypotheses, a non-peer-reviewed journal, by parents who launched the parental advocacy group SafeMinds to promote the theory that thiomersal caused autism. The controversy began to gain legitimacy in the eyes of the public and gained widening support within certain elements in the autism advocacy community as well as in the political arena, with U.S. Representative Dan Burton openly supporting this movement and holding a number of Congressional hearings on the subject.

Further support for the association between autism and thiomersal appeared in an article by Robert F. Kennedy Jr. in the magazines Rolling Stone and Salon.com alleging a government conspiracy at a CDC meeting to conceal the dangers of thiomersal to protect the pharmaceutical industry, and a book written by David Kirby, Evidence of Harm, dramatizing the lives of parents of autistic children, with both authors participating in media interviews to promote their work and the controversy. Although the allegations by Kennedy were denied and a US Senate committee investigation later found no evidence to substantiate the most serious allegations, the story had already been well publicized by leveraging Kennedy's celebrity. Salon magazine subsequently amended Kennedy's article five times due to factual errors and later retracted it completely on 16 January 2011, stating that the works of critics of the article and evidence of the flaws in the science connecting autism and vaccines undermined the value of the article to the editors.

Meanwhile, during this time of increased media publicity of the controversy, public health officials and institutions did little to rebut the concerns and speculative theories being offered. Media attention and polarization of the debate has also been fueled by personal injury lawyers who took out full-page ads in prominent newspapers and offered financial support for expert witnesses who dissented from the scientific consensus that there is no convincing evidence for a link between thiomersal and autism. Paul Offit, a leading vaccine researcher and advocate, has said that the media has a tendency to provide false balance by perpetually presenting both sides of an issue even when only one side is supported by the evidence and thereby giving a platform for the spread of misinformation.

Despite the consensus from experts that there is no link between thiomersal and autism, many parents continue to believe that such a link exists. These parents share the viewpoint that autism is not just treatable, but curable through "biomedical" interventions and have been frustrated by the lack of progress from more "mainline" scientists in finding this cure. Instead, they have supported an alternative community of like-minded parents, physicians and scientists who promote this belief. This mindset has taught these parents to challenge the expertise from the mainstream scientific community. Parents have also been influenced by an extensive network of anti-vaccination organizations such as Robert F. Kennedy Jr.'s Children's Health Defense and a large number of online anti-vaccination websites that present themselves as an alternative source for evidence using pseudoscientific claims. These websites use emotional appeals to gather support and frame the controversy as an adversarial dispute between parents and a conspiracy of doctors and scientists. Advocates for a thiomersal-autism link have also relied on celebrities like model Jenny McCarthy and information presented on Don Imus' Imus in the Morning radio show to persuade the public to their cause, instead of relying only on "dry" scientific papers and scientists. McCarthy has published a book describing her personal experience with her autistic son and appeared on The Oprah Winfrey Show to promote the hypothesis of vaccines causing autism. Bitterness over this issue has led to numerous threats made against the CDC as well as researchers like Offit, with increased security placed by the CDC in response to these threats.

Scientific evaluation

Rationale for doubting link

Various lines of evidence undermine a proposed link between thiomersal and autism. For example, although advocates of a thiomersal-autism link consider autism a form of "mercury poisoning," the typical symptoms of mercury toxicity are significantly different from symptoms seen in autism. Likewise, the neuroanatomic and histopathologic features of the brains of patients who have mercury poisoning, both with methylmercury as well as ethylmercury, have significant differences from the brains of people with autism. Previous episodes of widespread mercury toxicity in a population such as in Minamata Bay, Japan would also be expected to lead to documentation of a significant rise in autism or autism-like behavior in children should autism be caused by mercury poisoning. However, research on several episodes of acute and chronic mercury poisoning have not documented any such rise in autism-like behavior. Although some parents cite an association between the timing of onset of autistic symptoms with the timing of vaccinations as evidence of an environmental cause such as thiomersal, this line of reasoning can be misleading. Associations such as these do not establish causation as the two occurring together may be only coincidental in nature. Also, genetic disorders that have no environmental triggers such as Rett syndrome and Huntington's disease nevertheless have specific ages when they begin to show symptoms, suggesting specific ages of onset of symptoms does not necessarily require an environmental cause.

Although the concern for a thiomersal-autism link was originally derived from indirect evidence based on the known potent neurotoxic effects of methylmercury, recent studies show these feared effects were likely overestimated. Ethylmercury, such as in thiomersal, clears much faster from the body after administration than methylmercury, suggesting total mercury exposure over time is much less with ethylmercury. Currently used methods of estimating brain deposition of mercury likely overestimates the amounts deposited due to ethylmercury, and ethylmercury also decomposes quicker in the brain than methylmercury, suggesting a lower risk of brain damage. These findings show that the assumptions that originally led to concern about the toxicity of ethylmercury, which were based on direct comparison to methylmercury, were flawed.

Population studies

Multiple studies have been performed on data from large populations of children to study the relationship between the use of vaccines containing thiomersal, and autism and other neurodevelopmental disorders. Almost all of these studies have found no association between thiomersal-containing vaccines (TCVs) and autism, and studies done after the removal of thiomersal from vaccines have nevertheless shown autism rates continuing to increase. The only epidemiologic research that has found a purported link between TCVs and autism has been conducted by Mark Geier, whose flawed research has not been given any weight by independent reviews.

In Europe, a cohort study of 467,450 Danish children found no association between TCVs and autism or autism spectrum disorders (ASDs), nor any dose-response relationship between thiomersal and ASDs that would be suggestive of toxic exposure. An ecological analysis that studied 956 Danish children diagnosed with autism likewise did not show an association between autism and thiomersal. A retrospective cohort study on 109,863 children in the United Kingdom found no association between TCVs and autism, but a possible increased risk for tics. Analysis in this study also showed a possible protective effect with respect to general developmental disorders, attention-deficit disorder, and otherwise unspecified developmental delay. Another UK study based on a prospective cohort of 13,617 children likewise found more associated benefits than risks from thiomersal exposure with respect to developmental disorders. Because the Danish and UK studies involved only diphtheria-tetanus-pertussis (DTP) or diphtheria-tetanus (DT) vaccines, they are less relevant for the higher thiomersal exposure levels that occurred in the U.S.

In North America, a Canadian study of 27,749 children in Quebec showed that thiomersal was unrelated to the increasing trend in pervasive developmental disorders (PDDs). In fact, the study noted that rates of PDDs were higher in the birth cohorts with no thiomersal when compared to those with medium or high levels of exposure. A study performed in the US which analyzed data from 78,829 children enrolled in HMOs taken from the Vaccine Safety Datalink (VSD) did not show any consistent association between TCVs and neurodevelopmental outcomes, noting different results from data in different HMOs. A study performed in California found that removal of thiomersal from vaccines did not decrease the rates of autism, suggesting that thiomersal could not be the primary cause of autism. A study on children from Denmark, Sweden and California likewise argued against TCVs being causally associated with autism.

Scientific consensus

In 2001 the Centers for Disease Control and Prevention and the National Institutes of Health asked the U.S. National Academy of Sciences' (NAS) Institute of Medicine to establish an independent expert committee to review hypotheses about existing and emerging immunization safety concerns. This initial report found that based on indirect and incomplete evidence available at the time, there was inadequate evidence to accept or reject a thiomersal-autism link, though it was biologically plausible.

Since this report was released, several independent reviews have examined the body of published research for a possible thiomersal-autism link by examining the theoretical mechanisms of thiomersal causing harm and by reviewing the in vitro, animal, and population studies that have been published. These reviews determined that no evidence exists to establish thiomersal as the cause of autism or other neurodevelopmental disorders.

The scientific consensus on the subject is reflected in a follow-up report that was subsequently published in 2004 by the Institute of Medicine, which took into account new data that had been published since the 2001 report. The committee noted, in response to those who cite in vitro or animal models as evidence for the link between autism and thiomersal:

However, the experiments showing effects of thimerosal on biochemical pathways in cell culture systems and showing abnormalities in the immune system or metal metabolism in people with autism are provocative; the autism research community should consider the appropriate composition of the autism research portfolio with some of these new findings in mind. However, these experiments do not provide evidence of a relationship between vaccines or thimerosal and autism. In the absence of experimental or human evidence that vaccination (either the MMR vaccine or the preservative thimerosal) affects metabolic, developmental, immune, or other physiological or molecular mechanisms that are causally related to the development of autism, the committee concludes that the hypotheses generated to date are theoretical only.

The committee concludes:

Thus, based on this body of evidence, the committee concludes that the evidence favors rejection of a causal relationship between thimerosal-containing vaccines and autism.

Further evidence of the scientific consensus includes the rejection of a causal link between thiomersal and autism by multiple national and international scientific and medical bodies including the American Medical Association, the American Academy of Pediatrics, the American College of Medical Toxicology, the Canadian Paediatric Society, the U.S. National Academy of Sciences, the Food and Drug Administration, Centers for Disease Control and Prevention, the World Health Organization, the Public Health Agency of Canada, and the European Medicines Agency.

A 2011 journal article reflects this point of view and described the vaccine-autism connection as "the most damaging medical hoax of the last 100 years".

Consequences

The suggestion that thiomersal has contributed to autism and other neurodevelopmental disorders has had a number of effects. Public health officials believe fear driven by advocates of a thiomersal-autism link has caused parents to avoid vaccination or adopt "made up" vaccination schedules that expose their children to increased risk from preventable diseases such as measles and pertussis. Advocates of a thiomersal-autism link have also helped enact laws in six states (California, Delaware, Illinois, Missouri, New York and Washington) between 2004 and 2006 to limit the use of thiomersal given to pregnant women and children, although later attempts in 2009 in twelve other states failed to pass. These laws can be temporarily suspended, but vaccine advocates doubt their utility given the lack of evidence for danger with thiomersal in vaccines. Vaccine advocates are also concerned that passage of such laws help fuel a backlash against vaccination and contribute to doubts about the safety of vaccines that are unwarranted.

During the period of time of removal of thiomersal, the CDC and AAP asked doctors to delay the birth dose of hepatitis B vaccine in children not at risk for hepatitis. This decision, though following the precautionary principle, nevertheless sparked confusion, controversy and some harm. Approximately 10% of hospitals suspended the use of hepatitis B vaccine for all newborns, and one child born to a Michigan mother infected with hepatitis B virus died of it. Similarly, a study found that the number of hospitals who failed to properly vaccinate infants of hepatitis B seropositive mothers rose by over 6 times. This is a potential negative outcome given the high probability that infants who acquire hepatitis B infection at birth will develop the infection in a chronic form and possibly liver cancer.

The notion that thiomersal causes autism has led some parents to have their children treated with costly and potentially dangerous therapies such as chelation therapy, which is typically used to treat heavy metal poisoning, due to parental fears that autism is a form of "mercury poisoning". As many as 2 to 8% of autistic children in the U.S., numbering as many as several thousand children per year, receive mercury-chelating agents. Although critics of using chelation therapy as an autism treatment point to a lack of any evidence to support its use, hundreds of doctors prescribe these medications despite possible side effects including nutritional deficiencies as well as damage to the liver and kidney. The popularity of this therapy caused a "public health imperative" that led the U.S. National Institute of Mental Health (NIMH) to commission a study about chelation in autism by studying DMSA, a chelating agent used for lead poisoning, despite worries from critics that there would be no chance it would show positive results and it would be unlikely to convince parents to not use the therapy. Ultimately, the study was halted due to ethical concerns that there would be too much risk to children with autism who did not have toxic levels of mercury or lead due to a new animal study showing possible cognitive and emotional problems associated with DMSA. A 5-year-old autistic boy died from cardiac arrest immediately after receiving chelation therapy treatment using EDTA in 2005.

The notion has also diverted attention and resources away from efforts to determine the causes of autism.^[30] The 2004 Institute of Medicine report committee recommended that while it supported "targeted research that focuses on better understanding the disease of autism, from a public health perspective the committee does not consider a significant investment in studies of the theoretical vaccine-autism connection to be useful at this time." Alison Singer, a senior executive of Autism Speaks, resigned from the group in 2009 in a dispute over whether to fund more research on links between vaccination and autism, saying, "There isn't an unlimited pot of money, and every dollar spent looking where we know the answer isn't is one less dollar we have to spend where we might find new answers."

Court cases

Further information: Vaccine court

Further information: Autism omnibus trial

 

From 1988 until August 2010, 5,632 claims relating to autism were made to Office of Special Masters of the U.S. Court of Federal Claims (commonly known as the "Vaccine Court") which oversees vaccine injury claims, of which one case has received compensation, 738 cases have been dismissed with no compensations made, and with the remaining cases pending. In the one case which received compensation, the U.S. government agreed to pay for injury to a child that had a pre-existing mitochondrial disorder who developed autism-like symptoms after multiple vaccinations, some of which included thiomersal. Citing the inability to rule out a role of these vaccinations in exacerbating her underlying mitochondrial disorder as the rationale for payment, CDC officials cautioned against generalizing this one case to all autism-related vaccine cases as most patients with autism do not have a mitochondrial disorder. In February 2009, this court also ruled on three autism-related cases, each exploring different mechanisms that plaintiffs proposed linked thiomersal-containing vaccines with autism. Three judges independently found no evidence that vaccines caused autism and denied the plaintiffs compensation. Since these same mechanisms formed the basis for the vast majority of remaining autism-related vaccine injury cases, the chance for compensation in any of these cases has significantly decreased. In March 2010, the court ruled in three other test cases that thiomersal-containing vaccines do not cause autism.

Radio astronomy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Radio_astronomy 

The Karl G. Jansky Very Large Array, a radio interferometer in New Mexico, United States

Radio astronomy is a subfield of astronomy that studies celestial objects at radio frequencies. The first detection of radio waves from an astronomical object was in 1933, when Karl Jansky at Bell Telephone Laboratories reported radiation coming from the Milky Way. Subsequent observations have identified a number of different sources of radio emission. These include stars and galaxies, as well as entirely new classes of objects, such as radio galaxies, quasars, pulsars, and masers. The discovery of the cosmic microwave background radiation, regarded as evidence for the Big Bang theory, was made through radio astronomy.

Radio astronomy is conducted using large radio antennas referred to as radio telescopes, that are either used singularly, or with multiple linked telescopes utilizing the techniques of radio interferometry and aperture synthesis. The use of interferometry allows radio astronomy to achieve high angular resolution, as the resolving power of an interferometer is set by the distance between its components, rather than the size of its components.

Radio astronomy differs from radar astronomy in that the former is a passive observation (i.e., receiving only) and the latter an active one (transmitting and receiving).

History

Karl Jansky and his rotating directional antenna (early 1930s) in Holmdel, New Jersey, the world's first radio telescope, which was used to discover radio emissions from the Milky Way

Before Jansky observed the Milky Way in the 1930s, physicists speculated that radio waves could be observed from astronomical sources. In the 1860s, James Clerk Maxwell's equations had shown that electromagnetic radiation is associated with electricity and magnetism, and could exist at any wavelength. Several attempts were made to detect radio emission from the Sun including an experiment by German astrophysicists Johannes Wilsing and Julius Scheiner in 1896 and a centimeter wave radiation apparatus set up by Oliver Lodge between 1897 and 1900. These attempts were unable to detect any emission due to technical limitations of the instruments. The discovery of the radio reflecting ionosphere in 1902, led physicists to conclude that the layer would bounce any astronomical radio transmission back into space, making them undetectable.

Karl Jansky made the discovery of the first astronomical radio source serendipitously in the early 1930s. As a newly hired radio engineer with Bell Telephone Laboratories, he was assigned the task to investigate static that might interfere with short wave transatlantic voice transmissions. Using a large directional antenna, Jansky noticed that his analog pen-and-paper recording system kept recording a persistent repeating signal or "hiss" of unknown origin. Since the signal peaked about every 24 hours, Jansky first suspected the source of the interference was the Sun crossing the view of his directional antenna. Continued analysis, however, showed that the source was not following the 24-hour daily cycle of the Sun exactly, but instead repeating on a cycle of 23 hours and 56 minutes. Jansky discussed the puzzling phenomena with his friend, astrophysicist Albert Melvin Skellett, who pointed out that the observed time between the signal peaks was the exact length of a sidereal day; the time it took for "fixed" astronomical objects, such as a star, to pass in front of the antenna every time the Earth rotated. By comparing his observations with optical astronomical maps, Jansky eventually concluded that the radiation source peaked when his antenna was aimed at the densest part of the Milky Way in the constellation of Sagittarius.

Jansky announced his discovery at a meeting in Washington, D.C., in April 1933 and the field of radio astronomy was born. In October 1933, his discovery was published in a journal article entitled "Electrical disturbances apparently of extraterrestrial origin" in the Proceedings of the Institute of Radio Engineers. Jansky concluded that since the Sun (and therefore other stars) were not large emitters of radio noise, the strange radio interference may be generated by interstellar gas and dust in the galaxy, in particular, by "thermal agitation of charged particles." (Jansky's peak radio source, one of the brightest in the sky, was designated Sagittarius A in the 1950s and was later hypothesized to be emitted by electrons in a strong magnetic field. Current thinking is that these are ions in orbit around a massive Black hole at the center of the galaxy at a point now designated as Sagittarius A*. The asterisk indicates that the particles at Sagittarius A are ionized.)

After 1935, Jansky wanted to investigate the radio waves from the Milky Way in further detail, but Bell Labs reassigned him to another project, so he did no further work in the field of astronomy. His pioneering efforts in the field of radio astronomy have been recognized by the naming of the fundamental unit of flux density, the jansky (Jy), after him.

Grote Reber's Antenna at Wheaton, Illinois, world's first parabolic radio telescope

Grote Reber was inspired by Jansky's work, and built a parabolic radio telescope 9m in diameter in his backyard in 1937. He began by repeating Jansky's observations, and then conducted the first sky survey in the radio frequencies. On February 27, 1942, James Stanley Hey, a British Army research officer, made the first detection of radio waves emitted by the Sun. Later that year George Clark Southworth, at Bell Labs like Jansky, also detected radiowaves from the Sun. Both researchers were bound by wartime security surrounding radar, so Reber, who was not, published his 1944 findings first. Several other people independently discovered solar radio waves, including E. Schott in Denmark and Elizabeth Alexander working on Norfolk Island.

Chart on which Jocelyn Bell Burnell first recognised evidence of a pulsar, in 1967 (exhibited at Cambridge University Library)

At Cambridge University, where ionospheric research had taken place during World War II, J. A. Ratcliffe along with other members of the Telecommunications Research Establishment that had carried out wartime research into radar, created a radiophysics group at the university where radio wave emissions from the Sun were observed and studied. This early research soon branched out into the observation of other celestial radio sources and interferometry techniques were pioneered to isolate the angular source of the detected emissions. Martin Ryle and Antony Hewish at the Cavendish Astrophysics Group developed the technique of Earth-rotation aperture synthesis. The radio astronomy group in Cambridge went on to found the Mullard Radio Astronomy Observatory near Cambridge in the 1950s. During the late 1960s and early 1970s, as computers (such as the Titan) became capable of handling the computationally intensive Fourier transform inversions required, they used aperture synthesis to create a 'One-Mile' and later a '5 km' effective aperture using the One-Mile and Ryle telescopes, respectively. They used the Cambridge Interferometer to map the radio sky, producing the Second (2C) and Third (3C) Cambridge Catalogues of Radio Sources.

Techniques

Window of radio waves observable from Earth, on rough plot of Earth's atmospheric absorption and scattering (or opacity) of various wavelengths of electromagnetic radiation

Radio astronomers use different techniques to observe objects in the radio spectrum. Instruments may simply be pointed at an energetic radio source to analyze its emission. To "image" a region of the sky in more detail, multiple overlapping scans can be recorded and pieced together in a mosaic image. The type of instrument used depends on the strength of the signal and the amount of detail needed.

Observations from the Earth's surface are limited to wavelengths that can pass through the atmosphere. At low frequencies or long wavelengths, transmission is limited by the ionosphere, which reflects waves with frequencies less than its characteristic plasma frequency. Water vapor interferes with radio astronomy at higher frequencies, which has led to building radio observatories that conduct observations at millimeter wavelengths at very high and dry sites, in order to minimize the water vapor content in the line of sight. Finally, transmitting devices on Earth may cause radio-frequency interference. Because of this, many radio observatories are built at remote places.

Radio telescopes

Main article: Radio telescope

Radio telescopes may need to be extremely large in order to receive signals with low signal-to-noise ratio. Also since angular resolution is a function of the diameter of the "objective" in proportion to the wavelength of the electromagnetic radiation being observed, radio telescopes have to be much larger in comparison to their optical counterparts. For example, a 1-meter diameter optical telescope is two million times bigger than the wavelength of light observed giving it a resolution of roughly 0.3 arc seconds, whereas a radio telescope "dish" many times that size may, depending on the wavelength observed, only be able to resolve an object the size of the full moon (30 minutes of arc).

Radio interferometry

Main article: Astronomical interferometry

See also: Radio telescope § Radio interferometry

The Atacama Large Millimeter Array (ALMA), many antennas linked together in a radio interferometer

An optical image of the galaxy M87 (HST), a radio image of same galaxy using Interferometry (Very Large Array – VLA), and an image of the center section (VLBA) using a Very Long Baseline Array (Global VLBI) consisting of antennas in the US, Germany, Italy, Finland, Sweden and Spain. The jet of particles is suspected to be powered by a black hole in the center of the galaxy.

The difficulty in achieving high resolutions with single radio telescopes led to radio interferometry, developed by British radio astronomer Martin Ryle and Australian engineer, radiophysicist, and radio astronomer Joseph Lade Pawsey and Ruby Payne-Scott in 1946. The first use of a radio interferometer for an astronomical observation was carried out by Payne-Scott, Pawsey and Lindsay McCready on 26 January 1946 using a single converted radar antenna (broadside array) at 200 MHz near Sydney, Australia. This group used the principle of a sea-cliff interferometer in which the antenna (formerly a World War II radar) observed the Sun at sunrise with interference arising from the direct radiation from the Sun and the reflected radiation from the sea. With this baseline of almost 200 meters, the authors determined that the solar radiation during the burst phase was much smaller than the solar disk and arose from a region associated with a large sunspot group. The Australia group laid out the principles of aperture synthesis in a ground-breaking paper published in 1947. The use of a sea-cliff interferometer had been demonstrated by numerous groups in Australia, Iran and the UK during World War II, who had observed interference fringes (the direct radar return radiation and the reflected signal from the sea) from incoming aircraft.

The Cambridge group of Ryle and Vonberg observed the Sun at 175 MHz for the first time in mid July 1946 with a Michelson interferometer consisting of two radio antennas with spacings of some tens of meters up to 240 meters. They showed that the radio radiation was smaller than 10 arc minutes in size and also detected circular polarization in the Type I bursts. Two other groups had also detected circular polarization at about the same time (David Martyn in Australia and Edward Appleton with James Stanley Hey in the UK).

Modern radio interferometers consist of widely separated radio telescopes observing the same object that are connected together using coaxial cable, waveguide, optical fiber, or other type of transmission line. This not only increases the total signal collected, it can also be used in a process called aperture synthesis to vastly increase resolution. This technique works by superposing ("interfering") the signal waves from the different telescopes on the principle that waves that coincide with the same phase will add to each other while two waves that have opposite phases will cancel each other out. This creates a combined telescope that is the size of the antennas furthest apart in the array. In order to produce a high quality image, a large number of different separations between different telescopes are required (the projected separation between any two telescopes as seen from the radio source is called a "baseline") – as many different baselines as possible are required in order to get a good quality image. For example, the Very Large Array has 27 telescopes giving 351 independent baselines at once.

Very-long-baseline interferometry

Main article: Very-long-baseline interferometry

Beginning in the 1970s, improvements in the stability of radio telescope receivers permitted telescopes from all over the world (and even in Earth orbit) to be combined to perform very-long-baseline interferometry. Instead of physically connecting the antennas, data received at each antenna is paired with timing information, usually from a local atomic clock, and then stored for later analysis on magnetic tape or hard disk. At that later time, the data is correlated with data from other antennas similarly recorded, to produce the resulting image. Using this method it is possible to synthesise an antenna that is effectively the size of the Earth. The large distances between the telescopes enable very high angular resolutions to be achieved, much greater in fact than in any other field of astronomy. At the highest frequencies, synthesised beams less than 1 milliarcsecond are possible.

The pre-eminent VLBI arrays operating today are the Very Long Baseline Array (with telescopes located across North America) and the European VLBI Network (telescopes in Europe, China, South Africa and Puerto Rico). Each array usually operates separately, but occasional projects are observed together producing increased sensitivity. This is referred to as Global VLBI. There are also a VLBI networks, operating in Australia and New Zealand called the LBA (Long Baseline Array), and arrays in Japan, China and South Korea which observe together to form the East-Asian VLBI Network (EAVN).

Since its inception, recording data onto hard media was the only way to bring the data recorded at each telescope together for later correlation. However, the availability today of worldwide, high-bandwidth networks makes it possible to do VLBI in real time. This technique (referred to as e-VLBI) was originally pioneered in Japan, and more recently adopted in Australia and in Europe by the EVN (European VLBI Network) who perform an increasing number of scientific e-VLBI projects per year.

Astronomical sources

Main article: Astronomical radio source

See also: Radio object with continuous optical spectrum

A radio image of the central region of the Milky Way galaxy. The arrow indicates a supernova remnant which is the location of a newly discovered transient, bursting low-frequency radio source GCRT J1745-3009.

Radio astronomy has led to substantial increases in astronomical knowledge, particularly with the discovery of several classes of new objects, including pulsars, quasars and radio galaxies. This is because radio astronomy allows us to see things that are not detectable in optical astronomy. Such objects represent some of the most extreme and energetic physical processes in the universe.

The cosmic microwave background radiation was also first detected using radio telescopes. However, radio telescopes have also been used to investigate objects much closer to home, including observations of the Sun and solar activity, and radar mapping of the planets.

Other sources include:

• Sun
• Jupiter
• Sagittarius A, the Galactic Center of the Milky Way, with one portion Sagittarius A* thought to be a radio wave emitting supermassive black hole
• Active galactic nuclei and pulsars have jets of charged particles which emit synchrotron radiation
• Merging galaxy clusters often show diffuse radio emission
• Supernova remnants can also show diffuse radio emission; pulsars are a type of supernova remnant that shows highly synchronous emission.
• The cosmic microwave background is blackbody radio/microwave emission

International regulation

Antenna 70 m of the Goldstone Deep Space Communications Complex, California

Antenna 110m of the Green Bank radio telescope, US

Radio astronomy service (also: radio astronomy radiocommunication service) is, according to Article 1.58 of the International Telecommunication Union's (ITU) Radio Regulations (RR), defined as "A radiocommunication service involving the use of radio astronomy". Subject of this radiocommunication service is to receive radio waves transmitted by astronomical or celestial objects.

Frequency allocation

The allocation of radio frequencies is provided according to Article 5 of the ITU Radio Regulations (edition 2012).

In order to improve harmonisation in spectrum utilisation, the majority of service-allocations stipulated in this document were incorporated in national Tables of Frequency Allocations and Utilisations which is with-in the responsibility of the appropriate national administration. The allocation might be primary, secondary, exclusive, and shared.

• primary allocation: is indicated by writing in capital letters (see example below)
• secondary allocation: is indicated by small letters
• exclusive or shared utilization: is within the responsibility of administrations

In line to the appropriate ITU Region the frequency bands are allocated (primary or secondary) to the radio astronomy service as follows.

Allocation to services
Region 1	Region 2	Region 3
13 360–13 410 kHz  FIXED       RADIO ASTRONOMY
25 550–25 650          RADIO ASTRONOMY
37.5–38.25 MHz  FIXED MOBILE Radio astronomy
322–328.6     FIXED MOBILE RADIO ASTRONOMY
406.1–410     FIXED MOBILE except aeronautical mobile RADIO ASTRONOMY
1 400–1 427   EARTH EXPLORATION-SATELLITE (passive) RADIO ASTRONOMY SPACE RESEARCH (passive)
1 610.6–1 613.8 MOBILE-SATELLITE (Earth-to-space) RADIO ASTRONOMY AERONAUTICAL RADIONAVIGATION	1 610.6–1 613.8 MOBILE-SATELLITE (Earth-to-space) RADIO ASTRONOMY AERONAUTICAL RADIONAVIGATION RADIODETERMINATION- SATELLITE (Earth-to-space)	1 610.6–1 613.8 MOBILE-SATELLITE (Earth-to-space) RADIO ASTRONOMY AERONAUTICAL RADIONAVIGATION Radiodetermination- satellite (Earth-to-space)
10.6–10.68 GHz   RADIO ASTRONOMY and other services
10.68–10.7           RADIO ASTRONOMY and other services
14.47–14.5           RADIO ASTRONOMY and other services
15.35–15.4           RADIO ASTRONOMY and other services
22.21–22.5           RADIO ASTRONOMY and other services
23.6–24                RADIO ASTRONOMY and other services
31.3–31.5             RADIO ASTRONOMY and other services