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Propagator

In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. Propagators may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function).

Non-relativistic propagators
In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a particle to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t).

Consider a system with Hamiltonian $H$. The Green's function G (fundamental solution) for the Schrödinger equation is a function
 * $$G(x, t; x', t') = \frac{1}{i\hbar} \Theta(t - t') K(x, t; x', t')$$

satisfying
 * $$\left( i\hbar \frac{\partial}{\partial t} - H_x \right) G(x, t; x', t') = \delta(x - x') \delta(t - t'),$$

where $H_{x}$ denotes the Hamiltonian written in terms of the $x$ coordinates, $δ(x)$ denotes the Dirac delta-function, $Θ(t)$ is the Heaviside step function and $K(x, t ;x′, t′)$ is the kernel of the above Schrödinger differential operator in the big parentheses. The term propagator is sometimes used in this context to refer to $G$, and sometimes to $K$. This article will use the term to refer to $K$ (see Duhamel's principle).

This propagator may also be written as the transition amplitude
 * $$K(x, t; x', t') = \big\langle x \big| \hat{U}(t, t') \big| x' \big\rangle,$$

where $Û(t, t′)$ is the unitary time-evolution operator for the system taking states at time $t′$ to states at time $t$. Note the initial condition enforced by $$\lim_{t \to t'} K(x, t; x', t') = \delta(x - x')$$.

The quantum-mechanical propagator may also be found by using a path integral:
 * $$K(x, t; x', t') = \int \exp \left[\frac{i}{\hbar} \int_{t'}^{t} L(\dot{q}, q, t) \, dt\right] D[q(t)],$$

where the boundary conditions of the path integral include $q(t) = x, q(t′) = x′$. Here $L$ denotes the Lagrangian of the system. The paths that are summed over move only forwards in time and are integrated with the differential $$D[q(t)]$$ following the path in time.

In non-relativistic quantum mechanics, the propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is specified by the equation
 * $$\psi(x, t) = \int_{-\infty}^\infty \psi(x', t') K(x, t; x', t') \, dx'.$$

If $K(x, t; x′, t′)$ only depends on the difference $x − x′$, this is a convolution of the initial wave function and the propagator.

Basic examples: propagator of free particle and harmonic oscillator
For a time-translationally invariant system, the propagator only depends on the time difference $t − t′$, so it may be rewritten as $$K(x, t; x', t') = K(x, x'; t - t').$$

The propagator of a one-dimensional free particle, obtainable from, e.g., the path integral, is then

Similarly, the propagator of a one-dimensional quantum harmonic oscillator is the Mehler kernel,

The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity, $$\begin{align} &\exp \left( -\frac{it}{\hbar} \left( \frac{1}{2m} \mathsf{p}^2 + \frac{1}{2} m\omega^2 \mathsf{x}^2 \right) \right) \\ &= \exp \left( -\frac{im\omega}{2\hbar} \mathsf{x}^2\tan\frac{\omega t}{2} \right) \exp \left( -\frac{i}{2m\omega \hbar}\mathsf{p}^2 \sin(\omega t) \right) \exp \left( -\frac{im\omega }{2\hbar} \mathsf{x}^2 \tan\frac{\omega t}{2} \right), \end{align}$$ valid for operators $$\mathsf{x}$$ and $$\mathsf{p}$$ satisfying the Heisenberg relation $$[\mathsf{x},\mathsf{p}] = i\hbar$$.

For the $N$-dimensional case, the propagator can be simply obtained by the product $$K(\vec{x}, \vec{x}'; t) = \prod_{q=1}^N K(x_q, x_q'; t).$$

Relativistic propagators
In relativistic quantum mechanics and quantum field theory the propagators are Lorentz-invariant. They give the amplitude for a particle to travel between two spacetime events.

Scalar propagator
In quantum field theory, the theory of a free (or non-interacting) scalar field is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes spin-zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.

Position space
The position space propagators are Green's functions for the Klein–Gordon equation. This means that they are functions $G(x, y)$ satisfying $$\left(\square_x + m^2\right) G(x, y) = -\delta(x - y),$$ where
 * $x, y$ are two points in Minkowski spacetime,
 * $$\square_x = \tfrac{\partial^2}{\partial t^2} - \nabla^2$$ is the d'Alembertian operator acting on the $x$ coordinates,
 * $δ(x − y)$ is the Dirac delta function.

(As typical in relativistic quantum field theory calculations, we use units where the speed of light $c$ and the reduced Planck constant $ħ$ are set to unity.)

We shall restrict attention to 4-dimensional Minkowski spacetime. We can perform a Fourier transform of the equation for the propagator, obtaining $$\left(-p^2 + m^2\right) G(p) = -1.$$

This equation can be inverted in the sense of distributions, noting that the equation $xf(x) = 1$ has the solution (see Sokhotski–Plemelj theorem) $$f(x) = \frac{1}{x \pm i\varepsilon} = \frac{1}{x} \mp i\pi\delta(x),$$ with $ε$ implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.

The solution is

where $$p(x - y) := p_0(x^0 - y^0) - \vec{p} \cdot (\vec{x} - \vec{y})$$ is the 4-vector inner product.

The different choices for how to deform the integration contour in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the $$p_0$$ integral.

The integrand then has two poles at $$p_0 = \pm \sqrt{\vec{p}^2 + m^2},$$ so different choices of how to avoid these lead to different propagators.

Retarded propagator

A contour going clockwise over both poles gives the causal retarded propagator. This is zero if $x-y$ is spacelike or $y$ is to the future of $x$, so it is zero if $x ⁰< y ⁰$.

This choice of contour is equivalent to calculating the limit, $$G_\text{ret}(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{(p_0+i\varepsilon)^2 - \vec{p}^2 - m^2} = -\frac{\Theta(x^0 - y^0)}{2\pi} \delta(\tau_{xy}^2) + \Theta(x^0 - y^0)\Theta(\tau_{xy}^2)\frac{m J_1(m \tau_{xy})}{4 \pi \tau_{xy}}.$$

Here $$\Theta (x) := \begin{cases} 1 & x \ge 0 \\ 0 & x < 0 \end{cases}$$ is the Heaviside step function, $$\tau_{xy}:= \sqrt{ (x^0 - y^0)^2 - (\vec{x} - \vec{y})^2}$$ is the proper time from $x$ to $y$, and $$J_1$$ is a Bessel function of the first kind. The propagator is non-zero only if $$y \prec x$$, i.e., $y$ causally precedes $x$, which, for Minkowski spacetime, means
 * $$y^0 \leq x^0$$ and $$\tau_{xy}^2 \geq 0 ~.$$

This expression can be related to the vacuum expectation value of the commutator of the free scalar field operator, $$G_\text{ret}(x,y) = -i \langle 0| \left[ \Phi(x), \Phi(y) \right] |0\rangle \Theta(x^0 - y^0)$$ where $$\left[\Phi(x), \Phi(y) \right] := \Phi(x) \Phi(y) - \Phi(y) \Phi(x)$$ is the commutator.

Advanced propagator

A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if $x-y$ is spacelike or if $y$ is to the past of $x$, so it is zero if $x ⁰> y ⁰$.

This choice of contour is equivalent to calculating the limit $$ G_\text{adv}(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2\pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{(p_0 - i\varepsilon)^2 - \vec{p}^2 - m^2} = -\frac{\Theta(y^0-x^0)}{2\pi}\delta(\tau_{xy}^2) + \Theta(y^0-x^0)\Theta(\tau_{xy}^2)\frac{m J_1(m \tau_{xy})}{4 \pi \tau_{xy}}. $$

This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field. In this case, $$G_\text{adv}(x,y) = i \langle 0|\left[ \Phi(x), \Phi(y) \right]|0\rangle \Theta(y^0 - x^0)~.$$

Feynman propagator

A contour going under the left pole and over the right pole gives the Feynman propagator, introduced by Richard Feynman in 1948.

This choice of contour is equivalent to calculating the limit $$G_F(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{p^2 - m^2 + i\varepsilon} = \begin{cases} -\frac{1}{4 \pi} \delta(\tau_{xy}^2) + \frac{m}{8 \pi \tau_{xy}} H_1^{(1)}(m \tau_{xy}) & \tau_{xy}^2 \geq 0 \\ -\frac{i m}{ 4 \pi^2 \sqrt{-\tau_{xy}^2}} K_1(m \sqrt{-\tau_{xy}^2}) & \tau_{xy}^2 < 0. \end{cases} $$

Here, $H_{1}^{(1)}$ is a Hankel function and $K_{1}$ is a modified Bessel function.

This expression can be derived directly from the field theory as the vacuum expectation value of the time-ordered product of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same, $$ \begin{align} G_F(x-y) & = -i \lang 0|T(\Phi(x) \Phi(y))|0 \rang \\[4pt] & = -i \left \lang 0| \left [\Theta(x^0 - y^0) \Phi(x)\Phi(y) + \Theta(y^0 - x^0) \Phi(y)\Phi(x) \right] |0 \right \rang. \end{align}$$

This expression is Lorentz invariant, as long as the field operators commute with one another when the points $x$ and $y$ are separated by a spacelike interval.

The usual derivation is to insert a complete set of single-particle momentum states between the fields with Lorentz covariant normalization, and then to show that the $Θ$ functions providing the causal time ordering may be obtained by a contour integral along the energy axis, if the integrand is as above (hence the infinitesimal imaginary part), to move the pole off the real line.

The propagator may also be derived using the path integral formulation of quantum theory.

Dirac propagator
Introduced by Paul Dirac in 1938.

Momentum space propagator
The Fourier transform of the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators.

They are often written with an explicit $ε$ term although this is understood to be a reminder about which integration contour is appropriate (see above). This $ε$ term is included to incorporate boundary conditions and causality (see below).

For a 4-momentum $p$ the causal and Feynman propagators in momentum space are:

 * $$\tilde{G}_\text{ret}(p) = \frac{1}{(p_0+i\varepsilon)^2 - \vec{p}^2 - m^2}$$
 * $$\tilde{G}_\text{adv}(p) = \frac{1}{(p_0-i\varepsilon)^2 - \vec{p}^2 - m^2}$$
 * $$\tilde{G}_F(p) = \frac{1}{p^2 - m^2 + i\varepsilon}. $$

For purposes of Feynman diagram calculations, it is usually convenient to write these with an additional overall factor of $i$ (conventions vary).

Faster than light?
The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is nonzero outside of the light cone, though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle travelling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages?

The answer is no: while in classical mechanics the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is commutators that determine which operators can affect one another.

So what does the spacelike part of the propagator represent? In QFT the vacuum is an active participant, and particle numbers and field values are related by an uncertainty principle; field values are uncertain even for particle number zero. There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field $Φ(x)$ if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an EPR correlation. Indeed, the propagator is often called a two-point correlation function for the free field.

Since, by the postulates of quantum field theory, all observable operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables.

Regarding virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-antiparticle pair that eventually disappears into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman's language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone. However, no signaling back in time is allowed.

Explanation using limits
This can be made clearer by writing the propagator in the following form for a massless particle: $$G^\varepsilon_F(x, y) = \frac{\varepsilon}{(x - y)^2 + i \varepsilon^2}.$$

This is the usual definition but normalised by a factor of $$\varepsilon$$. Then the rule is that one only takes the limit $$\varepsilon \to 0$$ at the end of a calculation.

One sees that $$G^\varepsilon_F(x, y) = \frac{1}{\varepsilon} \quad\text{if} (x - y)^2 = 0,$$ and $$\lim_{\varepsilon \to 0} G^\varepsilon_F(x, y) = 0 \quad\text{if} (x - y)^2 \neq 0.$$ Hence this means that a single massless particle will always stay on the light cone. It is also shown that the total probability for a photon at any time must be normalised by the reciprocal of the following factor: $$ \lim_{\varepsilon \to 0} \int |G^\varepsilon_F(0, x)|^2 \, dx^3 = \lim_{\varepsilon \to 0} \int \frac{\varepsilon^2}{(\mathbf{x}^2 - t^2)^2 + \varepsilon^4} \, dx^3 = 2 \pi^2 |t|. $$ We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.

Propagators in Feynman diagrams
The most common use of the propagator is in calculating probability amplitudes for particle interactions using Feynman diagrams. These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for every internal line, that is, every line that does not represent an incoming or outgoing particle in the initial or final state. It will also get a factor proportional to, and similar in form to, an interaction term in the theory's Lagrangian for every internal vertex where lines meet. These prescriptions are known as Feynman rules.

Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be off shell. In fact, since the propagator is obtained by inverting the wave equation, in general, it will have singularities on shell.

The energy carried by the particle in the propagator can even be negative. This can be interpreted simply as the case in which, instead of a particle going one way, its antiparticle is going the other way, and therefore carrying an opposing flow of positive energy. The propagator encompasses both possibilities. It does mean that one has to be careful about minus signs for the case of fermions, whose propagators are not even functions in the energy and momentum (see below).

Virtual particles conserve energy and momentum. However, since they can be off shell, wherever the diagram contains a closed loop, the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman diagram requires an integral over a continuum of possible energies and momenta. In general, these integrals of products of propagators can diverge, a situation that must be handled by the process of renormalization.

Spin $1/2$
If the particle possesses spin then its propagator is in general somewhat more complicated, as it will involve the particle's spin or polarization indices. The differential equation satisfied by the propagator for a spin $1/2$ particle is given by

 * $$(i\not\nabla' - m)S_F(x', x) = I_4\delta^4(x'-x),$$

where $I_{4}$ is the unit matrix in four dimensions, and employing the Feynman slash notation. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation, $$S_F(x', x) = \int\frac{d^4p}{(2\pi)^4}\exp{\left[-ip \cdot(x'-x)\right]}\tilde S_F(p),$$ the equation becomes

\begin{align} & (i \not \nabla' - m)\int\frac{d^4p}{(2\pi)^4}\tilde S_F(p)\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt] = {} & \int\frac{d^4p}{(2\pi)^4}(\not p - m)\tilde S_F(p)\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt] = {} & \int\frac{d^4p}{(2\pi)^4}I_4\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt] = {} & I_4\delta^4(x'-x), \end{align} $$

where on the right-hand side an integral representation of the four-dimensional delta function is used. Thus

 * $$(\not p - m I_4)\tilde S_F(p) = I_4.$$

By multiplying from the left with $$(\not p + m)$$ (dropping unit matrices from the notation) and using properties of the gamma matrices, $$\begin{align} \not p \not p & = \tfrac{1}{2}(\not p \not p + \not p \not p) \\[6pt] & = \tfrac{1}{2}(\gamma_\mu p^\mu \gamma_\nu p^\nu + \gamma_\nu p^\nu \gamma_\mu p^\mu) \\[6pt] & = \tfrac{1}{2}(\gamma_\mu \gamma_\nu + \gamma_\nu\gamma_\mu)p^\mu p^\nu \\[6pt] & = g_{\mu\nu}p^\mu p^\nu = p_\nu p^\nu = p^2, \end{align}$$

the momentum-space propagator used in Feynman diagrams for a Dirac field representing the electron in quantum electrodynamics is found to have form

 * $$ \tilde{S}_F(p) = \frac{(\not p + m)}{p^2 - m^2 + i \varepsilon} = \frac{(\gamma^\mu p_\mu + m)}{p^2 - m^2 + i \varepsilon}.$$

The $iε$ downstairs is a prescription for how to handle the poles in the complex $p_{0}$-plane. It automatically yields the Feynman contour of integration by shifting the poles appropriately. It is sometimes written

 * $$\tilde{S}_F(p) = {1 \over \gamma^\mu p_\mu - m + i\varepsilon} = {1 \over \not p - m + i\varepsilon} $$

for short. It should be remembered that this expression is just shorthand notation for $(γ_{μ}p^{μ} − m)^{−1}$. "One over matrix" is otherwise nonsensical. In position space one has $$S_F(x-y) = \int \frac{d^4 p}{(2\pi)^4} \, e^{-i p \cdot (x-y)} \frac{\gamma^\mu p_\mu + m}{p^2 - m^2 + i \varepsilon} = \left( \frac{\gamma^\mu (x-y)_\mu}{|x-y|^5} + \frac{m}{|x-y|^3} \right) J_1(m |x-y|).$$

This is related to the Feynman propagator by

 * $$S_F(x-y) = (i \not \partial + m) G_F(x-y)$$

where $$\not \partial := \gamma^\mu \partial_\mu$$.

Spin 1
The propagator for a gauge boson in a gauge theory depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg, the propagator for a photon is
 * $${-i g^{\mu\nu} \over p^2 + i\varepsilon }.$$

The general form with gauge parameter $λ$, up to overall sign and the factor of $$i$$, reads
 * $$ -i\frac{g^{\mu\nu} + \left(1-\frac{1}{\lambda}\right)\frac{p^\mu p^\nu}{p^2}}{p^2+i\varepsilon}.$$

The propagator for a massive vector field can be derived from the Stueckelberg Lagrangian. The general form with gauge parameter $λ$, up to overall sign and the factor of $$i$$, reads
 * $$ \frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{m^2}}{k^2-m^2+i\varepsilon}+\frac{\frac{k_\mu k_\nu}{m^2}}{k^2-\frac{m^2}{\lambda}+i\varepsilon}.$$

With these general forms one obtains the propagators in unitary gauge for $λ = 0$, the propagator in Feynman or 't Hooft gauge for $λ = 1$ and in Landau or Lorenz gauge for $λ = ∞$. There are also other notations where the gauge parameter is the inverse of $λ$, usually denoted $ξ$ (see $R_{ξ}$ gauges). The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter.

Unitary gauge:
 * $$\frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{m^2}}{k^2-m^2+i\varepsilon}.$$

Feynman ('t Hooft) gauge:
 * $$\frac{g_{\mu\nu}}{k^2-m^2+i\varepsilon}.$$

Landau (Lorenz) gauge:
 * $$\frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{k^2}}{k^2-m^2+i\varepsilon}.$$

Graviton propagator
The graviton propagator for Minkowski space in general relativity is $$G_{\alpha\beta~\mu\nu} = \frac{\mathcal{P}^2_{\alpha\beta~\mu\nu}}{k^2} - \frac{\mathcal{P}^0_s{}_{\alpha\beta~\mu\nu}}{2k^2} = \frac{g_{\alpha\mu} g_{\beta\nu}+ g_{\beta\mu}g_{\alpha\nu}- \frac{2}{D-2} g_{\mu\nu}g_{\alpha\beta}}{k^2},$$ where $$D$$ is the number of spacetime dimensions, $$\mathcal{P}^2$$ is the transverse and traceless spin-2 projection operator and $$\mathcal{P}^0_s$$ is a spin-0 scalar multiplet. The graviton propagator for (Anti) de Sitter space is $$G = \frac{\mathcal{P}^2}{2H^2-\Box} + \frac{\mathcal{P}^0_s}{2(\Box+4H^2)},$$ where $$H$$ is the Hubble constant. Note that upon taking the limit $$H \to 0$$ and $$\Box \to -k^2$$, the AdS propagator reduces to the Minkowski propagator.

Related singular functions
The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. We follow the notation in Bjorken and Drell. See also Bogolyubov and Shirkov (Appendix A). These functions are most simply defined in terms of the vacuum expectation value of products of field operators.

Pauli–Jordan function
The commutator of two scalar field operators defines the Pauli–Jordan function $$\Delta(x-y)$$ by

 * $$\langle 0 | \left[ \Phi(x),\Phi(y) \right] | 0 \rangle = i \, \Delta(x-y)$$

with
 * $$\,\Delta(x-y) = G_\text{ret} (x-y) - G_\text{adv}(x-y)$$

This satisfies
 * $$\Delta(x-y) = -\Delta(y-x)$$

and is zero if $$(x-y)^2 < 0$$.

Positive and negative frequency parts (cut propagators)
We can define the positive and negative frequency parts of $$\Delta(x-y)$$, sometimes called cut propagators, in a relativistically invariant way.

This allows us to define the positive frequency part:
 * $$\Delta_+(x-y) = \langle 0 | \Phi(x) \Phi(y) |0 \rangle, $$

and the negative frequency part:
 * $$\Delta_-(x-y) = \langle 0 | \Phi(y) \Phi(x) |0 \rangle. $$

These satisfy
 * $$\,i \Delta = \Delta_+ - \Delta_-$$

and
 * $$(\Box_x + m^2) \Delta_{\pm}(x-y) = 0.$$

Auxiliary function
The anti-commutator of two scalar field operators defines $$\Delta_1(x-y)$$ function by
 * $$\langle 0 | \left\{ \Phi(x),\Phi(y) \right\} | 0 \rangle = \Delta_1(x-y)$$

with
 * $$\,\Delta_1(x-y) = \Delta_+ (x-y) + \Delta_-(x-y).$$

This satisfies $$\,\Delta_1(x-y) = \Delta_1(y-x).$$

Green's functions for the Klein–Gordon equation
The retarded, advanced and Feynman propagators defined above are all Green's functions for the Klein–Gordon equation.

They are related to the singular functions by
 * $$G_\text{ret}(x-y) = \Delta(x-y) \Theta(x^0-y^0) $$
 * $$G_\text{adv}(x-y) = -\Delta(x-y) \Theta(y^0-x^0) $$
 * $$2 G_F(x-y) = -i \,\Delta_1(x-y) + \varepsilon(x^0 - y^0) \,\Delta(x-y) $$

where $$\varepsilon(x^0-y^0)$$ is the sign of $$x^0-y^0$$.

title: Propagator


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Institute for International Assistance and Solidarity

The Institute for International Assistance and Solidarity (IFIAS) is an international non-governmental organization with headquarters in Brussels, Belgium, working for the promotion of democracy, transparency and human rights.

It was registered under Belgian law as a non-profit organization in 1997, but its roots date from the late 1970s in Germany: There it was found in Bonn by former German chancellor Willy Brandt and Gerd Greune in 1979 to support democratic forces and establish exchange with civil society in the countries of the Warsaw Pact.

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Tommy Turrentine

Thomas Walter Turrentine, Jr. (April 22, 1928 – May 13, 1997) was a swing and hard bop trumpeter and composer who was active between the 1940s and the 1960s. He rarely worked as a bandleader, and was known for his work as a sideman with drummer Max Roach and his younger brother, the saxophonist Stanley Turrentine.

Biography
Born in Pittsburgh, Pennsylvania, Turrentine played in the bands of Benny Carter, Earl Bostic, Charles Mingus, Billy Eckstine, Dizzy Gillespie, and Count Basie. He later recorded with Sonny Clark, Lou Donaldson, and his brother Stanley's bands. His working relationship with Max Roach was spawned in part when he joined the Max Roach Quintet in the late 1950s. Turrentine was also adept on the piano at chord blockings and was a compositional exponent of Thelonious Monk's earlier chordal voicings. His bebop compositions combined a sophisticated and emotional fusion and poignant lyricism reminiscent of Benny Golson and with the passionate, spirited influence of the Clifford Brown/Max Roach Quintet.

While his brother had a successful career and recorded a number of albums over his lifetime, Tommy only recorded one album under his name before retiring in the 1960s. In the 1970s he lived on the ground floor of a brownstone with his wife, Jane on West 82nd Street in New York City, a street which during that period had a number of jazz luminaries living along its blocks between Broadway and Central Park, including Tommy Flanagan and Pharoah Sanders.

Tommy Turrentine was rarely active after the 1970s. In the summer of 1979, he was one of several star trumpeters (including John Faddis and others) who appeared at the Village Gate for an all-star tribute to Blue Mitchell who had died earlier that year. Turrentine also appeared on the 1989 album Blue Delight by keyboardist Sun Ra.

He died of cancer at the age of 69.

As leader

 * 1960: Tommy Turrentine (Time Records, 1960) - with Stanley Turrentine, trombonist Julian Priester, pianist Horace Parlan, bassist Bob Boswell, drummer Max Roach

As sideman
With Ahmed Abdul-Malik With Paul Chambers With Sonny Clark With Lou Donaldson With Booker Ervin With Dexter Gordon With Rufus Jones With Philly Joe Jones With Abbey Lincoln With Jackie McLean With Horace Parlan With John Patton With Max Roach With Archie Shepp With Sun Ra With Stanley Turrentine
 * The Music of Ahmed Abdul-Malik (New Jazz, 1961)
 * Sounds of Africa (New Jazz, 1961)
 * 1st Bassman (VeeJay, 1960)
 * Leapin' and Lopin' (Blue Note, 1961)
 * The Natural Soul (Blue Note, 1962)
 * Signifyin' (Argo, 1963)
 * The Book Cooks (Bethlehem, 1960)
 * Landslide (Blue Note, 1961-62 [1980])
 * Five on Eight (Cameo, 1964)
 * Mean What You Say (Sonet, 1977)
 * Abbey Is Blue (Riverside, 1959)
 * A Fickle Sonance (Blue Note, 1961)
 * Speakin' My Piece (Blue Note, 1960)
 * On the Spur of the Moment (Blue Note 1961)
 * Blue John (Blue Note, 1963)
 * Quiet as It's Kept (Mercury, 1959)
 * Moon Faced and Starry Eyed (Mercury, 1959)
 * Long as You're Living (Enja, 1960 [1984])
 * Parisian Sketches (Mercury, 1960)
 * Mama Too Tight (Impulse!, 1966)
 * Blue Delight (A&M, 1989)
 * Comin' Your Way (Blue Note, 1961)
 * Jubilee Shout!!! (Blue Note, 1962)
 * A Bluish Bag (Blue Note, 1967)
 * The Man with the Sad Face (Fantasy, 1976)

title: Tommy Turrentine


text: Category:Cyanides

title: Category:Cyanides


text:
Church

Church may refer to:

Religion

 * Church (building), a place/building for Christian religious activities and praying
 * Church (congregation), a local congregation of a Christian denomination
 * Church service, a formalized period of Christian communal worship
 * Christian denomination, a Christian organization with distinct doctrine and practice
 * Christian Church, either the collective body of all Christian believers, or early Christianity

United Kingdom

 * Church (Kensington and Chelsea ward), a former electoral ward of Kensington and Chelsea London Borough Council that existed from 1964 to 2002
 * Church (Liverpool ward), a Liverpool City Council ward
 * Church (Reading ward), a Reading Borough Council ward
 * Church (Sefton ward), a Metropolitan Borough of Sefton ward
 * Church, Lancashire, England

United States

 * Church, Iowa, an unincorporated community
 * Church Lake, a lake in Minnesota
 * Church, Michigan, ghost town

Arts, entertainment, and media

 * Church magazine, a pastoral theology magazine published by the National Pastoral Life Center

Fictional entities

 * Church (Red vs. Blue), a fictional character in the video web series Red vs. Blue
 * Church, a cat in Stephen King's novel Pet Sematary
 * Church, a meeting of SOA members in the FX television series Sons of Anarchy

Films

 * The Church (1989 film), an Italian horror film directed by Michele Soavi
 * The Church (2018 film), an American horror film directed by Dom Franklin

Groups

 * Chvrches, an electropop trio from Glasgow, Scotland
 * The Church (band), an Australian psychedelic rock band formed in Sydney in 1980

Albums

 * Church (album), a 2020 album by Swedish electronic dance music duo Galantis
 * The Church (Mr. Oizo album), a 2014 album by French electronic musician Mr. Oizo
 * The Church (The Church album), originally released in 1981 as Of Skins and Heart
 * Churches (album), a 2021 album by American singer-songwriter LP

Songs

 * "Church" (Alison Wonderland song)
 * "Church" (Coldplay song)
 * "Church" (T-Pain song)
 * "Church", a song by Aly & AJ from Sanctuary
 * "Church", a song by Galactic from Coolin' Off
 * "Church", a song by Fall Out Boy from Mania
 * "Church", a song by OutKast from Speakerboxxx/The Love Below
 * "Church", a 2021 song from Tina Arena
 * "The Church", a song by Hawkwind from Church of Hawkwind

Other uses

 * Church (programming language), a LISP-like probabilistic programming language
 * Church (surname), including a list of people
 * Church Line, San Francisco, California

title: Church


text:
User:Yogi

I have been the user of Wikipedia since mid 2003. My first contribution to the Wikipedia was creation of a new section for Spelling Bee.

title: User:Yogi


text: User talk:64.86.50.16

title: User talk:64.86.50.16


text:
Strachan Open

The Strachan Open was a professional snooker tournament held between 1992 and 1994. It had varying ranking status during its history.

History
The tournament was first held in 1992, and was a ranking tournament. It was sponsored by billiard cloth company Strachan. The event was boycotted by several top players due to lower points tariff and prize money. The final stages were played at the Thornbury Leisure Centre in Thornbury. Peter Ebdon made his first official maximum break in the qualifying rounds against Wayne Martin.

The following season to encourage the new professional, the World Professional Billiards and Snooker Association created four so-called minor ranking events. These tournaments were open to every professional, but earned only one-tenth of the ranking points of most other ranking tournaments. Three of the four minor ranking events were Strachan Challenges. The fourth was the Benson & Hedges Championship. Most of the top sixteen players did not bother to enter. The Strachan Challenges continued for one more season with two events, but they did not carry ranking points.

title: Strachan Open


text:
Talk:Longford (disambiguation)

2004
I propose moving this page to Longford (disambiguation), and moving Longford Town to this page, with a link at top of page.

Other Irish counties have the county at "County Name", with any county town of the same name at "Name".

A link to the disambiguation page should be sufficient for other uses, Longford Ireland is likely the most common usage.

Comments?

Zoney 14:03, 2 Jun 2004 (UTC)

Requested move 1 April 2019

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this section.

The result of the move request was: No consensus. Not moved. (non-admin closure) В²C ☎ 20:04, 15 April 2019 (UTC)

– Procedural nomination, I am neutral. While I favoured having the town at the base name serving as a broad concept article at Talk:Antrim, County Antrim and Talk:Donegal (town) the consensus in both of those cases was that the town was not primary. In the case of Antrim the town has a population of 20,001, and the county-618,108. With Antrim, New Hampshire only has 2,637 (the others don't have stated populations). In the case of Donegal the town has a population of 2,618 and the county 159,192. While Donegal, Pennsylvania only has 116 and is named after the Irish town. Donegal, Perth County, Ontario is also named after the Irish town and doesn't have a stated population. In both cases the towns of Antrim and Donegal could function as WP:DABCONCEPTS for the counties and thus would be by far the most populated/significant but there was consensus against that. In the case of Longford the town has a population of 10,008 and the county has 40,873. Longford, Coventry has 18,538, Longford, Derbyshire has 349, Longford, Gloucestershire 1,218, Longford (ward) has 12,163, Longford, Newport has 102, Longford, Warrington has 8,012, Longford, Victoria has 929, Longford, Tasmania has 3,863 and Longford, Kansas has 75 (total 45, 249). There is also Longford (film) and several other places such as Longford, London that don't have population data. Longford, County Longford and Longford Town F.C. have 6, 191 views combined while the others have 4,123. [|County_Longford|Longford,_Coventry|Longford,_Gloucestershire|Longford,_Derbyshire|Longford_(ward)|Longford_(film)|Longford,_Warrington|Longford,_Tasmania|Longford_Town_F.C.] [|Longford,_Newport|Longford,_Market_Drayton|Longford,_Victoria|Longford,_U.S._Virgin_Islands|Longford,_Ontario|Longford,_Wiltshire|Langford_(Wiltshire)_railway_station]. This isn't a WP:POINTy nomination but based on the previous consensus, if a town that has a county named after it and a few small places isn't primary then a town that has a county and some unrelated places that exceed even the county's population clearly isn't primary. I have chosen Longford, County Longford as the target like Category:Wexford, County Wexford due to the fact that Longford, Tasmania and Longford, Victoria are also towns and thus would be WP:PDABs and fail WP:PRECISION however I'm happy for Longford (town) to redirect to the Irish town or we could use Longford (Irish town). Commons:Category:Longford, County Longford is also the title of the Commons category.  Crouch, Swale  ( talk ) 13:26, 1 April 2019 (UTC)
 * Longford (disambiguation) → Longford
 * Longford → Longford, County Longford
 * Support. There are 49 entries at the Longford (disambiguation) page and a town of only 10,000, which does not have an iconic name, such as Hastings or Waterloo, cannot rise to the level of WP:PRIMARYTOPIC. —Roman Spinner (talk • contribs) 06:12, 2 April 2019 (UTC)
 * Support Longford (town). The county is much better known than the town, and like most Irish countries is frequently just known by its name without "County". But the Irish town is primary over the Australian town and a hatnote is fine to point to the latter. -- Necrothesp (talk) 15:18, 4 April 2019 (UTC)
 * I wouldn't use just "town" even though even though its clearly more important than the 2 Australian towns and Longford, Tasmania has 3,863 people compared to 10,008 for Ireland. I'd still say Longford, County Longford is the simplest disambiguation here though as noted Longford (Irish town) would work.  Crouch, Swale  ( talk ) 19:35, 5 April 2019 (UTC)
 * Comment: Longford, County Longford just looks horrible to me. Longford town/county tends to be differentiated myself by the context in which its used, but to me Longford would mean the town because of the railway stop there and because of use of Longford on road signage.  I'd be comfortable with Longford (town) but there's a dab issue, Longford (Irish town) would probably work OK for me.  Another option would be Longford Town, County Longford which somehow looks betters and distinguishes from the Longford (County Longford barony) might be another option.  I'm useless at titles so I'll just comment ... the changes usually give grief to my watchlist. ThankyouDjm-leighpark (talk) 19:56, 6 April 2019 (UTC)
 * Oppose: the town is both the county town and largest town in the county, by far. In everyday Ireland usage, "Longford" is synonymous with the town, the county would be made explicit. The alternative titles are mostly unwieldy, but more importantly would be attempting to fix a problem that I'm not sure exists in the first place. Declangi (talk) 02:04, 7 April 2019 (UTC)
 * Support; the town isn't the primary topic (I agree with Roman Spinner), so the problem that exists is that the title "Longford" is landing on a non-primary topic. Given that the town needs a qualified name, I believe that the county name is the usual qualifier. -- JHunterJ (talk) 13:51, 9 April 2019 (UTC)
 * Oppose per Declangi and general ugliness/issue with disambiguators. This RM seems like a solution looking for a problem to me. Leave as is. — Amakuru (talk) 10:19, 10 April 2019 (UTC)
 * Donegal and Antrim have less ambiguity issues than Longford yet the consensus was for disambiguation there, so it should definitely be here.  Crouch, Swale  ( talk ) 11:18, 10 April 2019 (UTC)
 * This oppose argument seems to be "the article isn't the primary topic, but we don't like the qualified title, therefore the article turns out the be the primary topic, so no problem!" The problem is the non-primaryness of the topic. The solution proposed is possibly not the optimal one. The approach should then be to find the optimal qualified title, not to ignore the problem, and certainly not to spin it as if the proposer and supporters are just creating problems for the proposal to fix. -- JHunterJ (talk) 12:18, 10 April 2019 (UTC)
 * I agree with Amakuru that the suitability of the target (including any alternatives) may be taken into account (particularly in borderline cases) but given the other 2 RMs with less ambiguous titles it would seem illogical to have this one as primary. In this case if there is consensus to move the closer should pick the best choice and then a new RM can be started for the end title (per WP:THREEOUTCOMES).  Crouch, Swale  ( talk ) 12:35, 10 April 2019 (UTC)
 * Neutral: Support: Longford (town) ... per Donegal (town) ... and I'd expect Wexford->Wexford (town) to go the same way especially with mention in Irish Soldier Laddie. Oppose all else. (Said would not oppose Sarah777)  The Antrim, County Antrim obviously took a different route but I notice we have  and to top it all .  Note:  I've left my comment above but come down from the fence after determining that for stay as we are ... resistance is futile.Djm-leighpark (talk) 12:23, 10 April 2019 (UTC)
 * Wouldn't Longford (Irish town) be better to avoid ambiguity with the 2 Australian towns.  Crouch, Swale  ( talk ) 12:35, 10 April 2019 (UTC)
 * I personally am less keen on the Irish town bit my preference being simply just town. Weirdly while applying banner to Wikivoyage and rechecking to ensure that was alright as Longford struck me as an usual place for a swimming pool and I couldn't recall one there I happened to notice said swimming pool] gives its address as Longford Town.  That exploration also indicated  has been there and might have an opinion and I'd likely not go against it.  In fact have WikiProject Ireland been directly consulted? Djm-leighpark (talk) 12:53, 10 April 2019 (UTC)
 * I'm not sure what's wrong with "(Irish town)" this is consistent with standard disambiguators such as Tyrone Power (Irish actor) and Georgia (U.S. state), while Newport (city), Vermont and Windermere, Cumbria (town) don't have this, Windermere (Cumbrian town) was suggested (and should probably end up being the title) and the ", Vermont" is used per WP:USPLACE.  Crouch, Swale  ( talk ) 06:30, 15 April 2019 (UTC)
 * Oppose - "Longford" for the town and "County Longford" for the county is the best solution. Leave it as is. Sarah777 (talk) 16:05, 10 April 2019 (UTC)
 * While I know googling isn't decisive it is notable that typing "Longford, Ireland" yields 10.4 million results while "Longford, Tasmania" gives 640,000. Sarah777 (talk) 16:22, 10 April 2019 (UTC)

 * Pinging the remaining participants at the Donegal discussion since we don't have consensus here.  Crouch, Swale  ( talk )  06:30, 15 April 2019 (UTC)
 * If we don't have consensus then we leave it as is. — Amakuru (talk) 06:55, 15 April 2019 (UTC)
 * Oppose - it's fine as it is. WP:OSE arguments should not bear on this discussion. Mjroots (talk) 07:02, 15 April 2019 (UTC)

 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Requested move 8 May 2021

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: No consensus to move. Many opposers argue that the Irish town is primary by long-term significance  (t · c)  buidhe  00:06, 16 May 2021 (UTC)

– I'm starting a new RM with a different target to 2 years ago. As noted previously the town has around 10,000 and the wider county around 40,000 but the other places combined have around 50,000. Also the film gets more views than the town[]. Alternative titles include Longford, Ireland (town) or as previously Longford, County Longford. Unlike the Bray move that created many links to the DAB I have DisamAssist so I can easily fix the links.  Crouch, Swale  ( talk ) 20:51, 8 May 2021 (UTC)
 * Longford (disambiguation) → Longford
 * Longford → Longford (Irish town)

 * Support – clearly a no primarytopic situation. No particular opinion whether Irish town is the best disambiguator, but it will do. Dicklyon (talk) 02:48, 9 May 2021 (UTC)
 * Support per nom.--Ortizesp (talk) 05:45, 9 May 2021 (UTC)
 * Oppose: The page view argument given over a one month period barely holds over a two year period and the maximum count for the town over a month was higher at circa 6500 versus 3000. A BOT analysis might also be interesting as they may be running film nav templates. I bring forward  comments of WP:OSE from the previous discussion, 's &  comments about the ugliness etc and 's previous !vote to leave as is. Thankyou. Djm-leighpark (talk) 21:09, 9 May 2021 (UTC)
 * Oppose - The problem we seem to be trying to solve (that the film has perhaps 6 more daily views than the town (129 v 123)) is a negligible one. And the solution proposed (move the article on the town to one with an atypical parenthetical disambiguation not common for geographic subjects) seems more than a little awkward. When the same problem is already solved by the simple DAB hatnote. As was noted by some of the contributors to the previous (2019) discussion, this seems like a solution looking for a problem. Where the cure is only marginally better than the disease. If at all. And the existing solution (the hatnote) already largely covers it anyway. Guliolopez (talk) 08:59, 10 May 2021 (UTC)
 * Oppose - per GulioLopez. Additionally, if we're talking page views, the town is clearly the major topic - it's been there for hundreds of years, it will be for hundreds more, while the film will fade into relative obscurity. Bastun Ėġáḍβáś₮ŭŃ! 11:33, 10 May 2021 (UTC)
 * Oppose - per Guliolopez. Nothing has changed in the last 2 years to warrant another discussion. Spleodrach (talk) 13:52, 10 May 2021 (UTC)
 * Oppose - per Guliolopez. ser! (chat to me - see my edits) 14:19, 10 May 2021 (UTC)
 * Oppose - "Irish town" is an awkward disambiguator. But I still don't see the need for any disambiguation. The popularity of a film waxes and wanes. None of the other Longford places come close to Ireland's Longford in prominence. And the 50,000 population of other places combined would seem to include the Longford ward of Coventry which, despite its population, seems very unlikely to be searched for as often as the Irish Longford. Declangi (talk) 06:38, 11 May 2021 (UTC)

title: Talk:Longford (disambiguation)


text:
Bile acid sequestrant

The bile acid sequestrants are a group of resins used to bind certain components of bile in the gastrointestinal tract. They disrupt the enterohepatic circulation of bile acids by combining with bile constituents and preventing their reabsorption from the gut. In general, they are classified as hypolipidemic agents, although they may be used for purposes other than lowering cholesterol. They are used in the treatment of chronic diarrhea due to bile acid malabsorption.

Mechanism
Bile acid sequestrants are polymeric compounds that serve as ion-exchange resins. Bile acid sequestrants exchange anions such as chloride ions for bile acids. By doing so, they bind bile acids and sequester them from the enterohepatic circulation. The liver then produces more bile acids to replace those that have been lost. Because the body uses cholesterol to make bile acids, this reduces the level of LDL cholesterol circulating in the blood.

Bile acid sequestrants are large polymeric structures, and they are not significantly absorbed from the gut into the bloodstream. Thus, bile acid sequestrants, along with any bile acids bound to the drug, are excreted via the feces after passage through the gastrointestinal tract.

Hyperlipidemia
As bile acids are biosynthesized from cholesterol, disruption of bile acid reabsorption will decrease cholesterol levels, in particular, low-density lipoprotein (commonly known as "bad cholesterol") in blood. Consequently, these drugs have been used for the treatment of hypercholesterolemia and dyslipidemia.

Use of these agents as hypolipidemic agents has decreased markedly since the introduction of the statins, which are more efficacious than bile acid sequestrants at lowering LDL. They are occasionally used as an adjunct to the statins as an alternative to the fibrates (another major group of cholesterol-lowering drugs), which are thought to increase the risk of rhabdomyolysis when used with statins. The bile-acid-binding resins can raise triglycerides modestly (about 5%) and cannot be used if the triglycerides are elevated.

Bile acid malabsorption
Chronic diarrhea may be caused by excess bile salts entering the colon rather than being absorbed at the end of the small intestine (the ileum). This condition of bile acid malabsorption occurs after surgery to the ileum, in Crohn's disease, with a number of other gastrointestinal causes, or is commonly a primary, idiopathic condition. The SeHCAT test can be used for diagnosis. Bile salt diarrhea can also be a side-effect of gallbladder removal.

Bile acid sequestrants are the principal therapy for bile acid-induced diarrhea. Cholestyramine, colestipol and colesevelam have all been used. Doses may not need to be as high as those previously used for hyperlipidemia. Many patients find them hard to tolerate, as although the diarrhea may improve, bloating and abdominal pain can worsen.

Use in other conditions
In chronic liver diseases such as cirrhosis, bile acids may deposit in the skin, causing pruritus (itching). Hence, bile acid sequestrants may be used for the prevention of pruritus in patients with chronic liver disease.

Bile acid sequestrants may also be used to treat hyperthyroidism as an adjunct therapy. By inhibiting the enterohepatic circulation, more L-thyroxine will be lost through defecation, thus lowering body thyroxine levels.

Cholestyramine has been used in the treatment of Clostridioides difficile infections, in order to absorb toxins A and B.

Side effects
As bile acid sequestrants are designed to stay in the gut; in general, they do not have systemic side effects. However, they may cause problems in the gastrointestinal tract, such as constipation, diarrhea, bloating, and flatulence. Some patients complain of the bad taste.

Because bile acid sequestrants are not well-absorbed from the gut, they are generally regarded as safe in pregnant women. However, by interfering with vitamin absorption (see below), they could cause vitamin deficiencies that may affect the fetus. So, vitamin supplementation may be considered, with appropriate intervals between dosing of the vitamins and bile acid sequestrants.

Drug interactions
In addition to bile acids, bile acid sequestrants may also bind drugs in the GI tract, preventing their absorption into the bloodstream. For this reason, it is generally advised that bile acid sequestrants be spaced several hours apart from other drugs.

Vitamins
They can also bind fat-soluble vitamins, such as vitamin A, vitamin D, vitamin E, and vitamin K. This effect could result in a vitamin deficiency, and so checking blood levels and possible supplementation has been suggested.

Examples
Three drugs are members of this class; all are synthetic polymeric resins:
 * Cholestyramine (generic and various proprietary names)
 * Colestipol (Colestid, Colestipid)
 * Colesevelam (Cholestagel in Europe, Welchol in the US, Lodalis in Canada)

title: Bile acid sequestrant


text:
FA Trophy

The Football Association Challenge Trophy, commonly known as the FA Trophy, is a men's football knockout cup competition run by and named after the English Football Association and competed for primarily by semi-professional teams. The competition was instigated in 1969 to cater to those non-league clubs that paid their players and were therefore not eligible to enter the FA Amateur Cup.

Eligibility rules have changed over time, but from 2008 onwards the competition has been open to clubs playing in Steps 1–4 of the National League System, equivalent to tiers 5–8 of the overall English football league system. This covers the National League, the Southern League, Isthmian League, and Northern Premier League.

The final of the competition was held at the original Wembley Stadium from the tournament's instigation until the stadium closed in 2000. The final has been played at the new Wembley Stadium since its opening in 2007. The record for the most FA Trophy wins is shared by Woking and two defunct clubs, Scarborough and Telford United, with three victories each.

The Trophy is currently held by Gateshead F.C, who beat Solihull Moors in the 2024 final held on 11 May 2024.

History
The competition was created by the Football Association in 1969 to afford semi-professional teams an opportunity to compete for the chance to play at the Wembley Stadium. Fully-amateur clubs took part in the long-standing FA Amateur Cup, but most of the leading non-league clubs made at least some form of payment to their players and were therefore ineligible to enter the Amateur Cup. The first winners of the competition were Macclesfield Town of the Northern Premier League, who defeated Telford United of the Southern League in the final. Northern Premier League clubs dominated the first decade of the competition, with Telford United the only Southern League team to break the northern clubs' hold on the competition. In the early years of its existence, the competition struggled to achieve the same level of prestige as the long-established Amateur Cup. In 1974 the FA abolished the distinction between official professional and amateur status and discontinued the Amateur Cup, and the Trophy soon had 300 entrants. This figure was gradually reduced until by 1991 only around 120 clubs took part. In 1978 the FA moved the final of the Trophy to the Saturday immediately following the FA Cup Final, so as to give it a longer build-up and avoid conflict with clubs' league programmes, which had previously reduced the competition's prestige.

In 1979 the leading Southern and Northern Premier League teams formed the new Alliance Premier League, and teams from this league dominated the Trophy during the 1980s, although in the 1980–81 season Bishop's Stortford of the comparatively lowly Isthmian League First Division entered at the preliminary round and won twelve matches to reach the final, where they defeated Sutton United. Telford United's win in 1989 made them the second team to win the Trophy three times. Between 1990 and 2000 three more teams claimed multiple wins. Former Northern Ireland international Martin O'Neill, in his third managerial role, led Wycombe Wanderers to two wins, and Geoff Chapple managed Kingstonian to victory twice and Woking three times, all within the space of seven years. After Chapple's period of success, Mark Stimson became the first man to manage the Trophy-winning team in three successive seasons, when he led Grays Athletic to victory in 2005 and 2006 and repeated the feat with his new club Stevenage Borough in 2007. Originally the competition included as many qualifying rounds as were required to reduce the number of teams to 32. In 1999 the format was amended to match that of the FA Cup, with six rounds prior to the semi-final stage, albeit without qualifying rounds. Teams from the Football Conference received byes through the early rounds, in a similar manner to the way in which the leading clubs receive byes in the FA Cup. As of 2001 the competition was sponsored by Umbro; in the 2007-08 season it was sponsored by Carlsberg.

Format
The competition is a knockout tournament with pairings drawn at random. Since the 2021–22 season all drawn games are immediately settled via a penalty shootout. This was introduced to ease fixture congestion.

The Semi-Finals follow the same procedure as the preceding rounds: hosted by the team drawn first out of the hat and decided by a penalty shootout if the scores remain level after 90 minutes.

The Final is held at Wembley Stadium. Thirty minutes of extra-time is played if the scores are level at full-time. A penalty shootout is used if the scores further remain level at the end of extra-time.

As of 2022–23 and the recent non–league restructuring, the competition has featured a maximum of 320 teams representing all clubs from levels 5-8 on the English football league system. The competition features three qualifying rounds (three less than the FA Cup) and eight rounds proper (the same as the FA Cup). Teams from Step 4 (level 8) enter at the first qualifying round stage, those from Step 3 (level 7) at the third qualifying, those from Step 2 (level 6) at the second round proper, and those from Step 1 (level 5) at the third round proper (similar to the FA Cup).

The months in which rounds are played are typical, with exact dates subject to each calendar. The FA pays a cumulative prize fund that rises round-by-round, with exact figures subject to each new edition. See the list of FA Trophy seasons for specific details.

Venues
The final was traditionally held at the original Wembley Stadium, but was moved to Villa Park during Wembley's redevelopment, and a final was also played at West Ham United's Boleyn Ground. Three replays were required, two of which were held at The Hawthorns, home of West Bromwich Albion, and one at Stoke City F.C.'s Victoria Ground.

In 2007 the final moved to the new Wembley Stadium, and a record crowd of 53,262 saw Stevenage Borough beat Kidderminster Harriers.

FA Trophy winners and finalists
Scarborough (1973, 1976, 1977), Telford United (1971, 1983, 1989), and Woking (1994, 1995, 1997) share the record for the most victories (three) in the final. In 1985 Wealdstone became the first team to win the "Non-League Double" of FA Trophy and Football Conference championship (although in the pre-Conference era both Macclesfield Town and Stafford Rangers had done the double of Northern Premier League championship and FA Trophy in 1970 and 1972 respectively). Since then Colchester United in 1992 and Wycombe Wanderers in 1993 have equalled Wealdstone's achievement.

Three Welsh clubs have reached FA Trophy finals; Bangor City (1984), Newport County (2012), and Wrexham in 2013, 2015 and 2022. Wrexham became the first Welsh winners of the FA Trophy in 2013.

Media coverage
From the 2004–05 season Sky Sports had a deal to show the final of the FA Trophy. This changed in 2007 when the FA agreed a new deal with Setanta Sports to provide coverage of FA Trophy matches with effect from the 2008–09 season. In March 2013 it was announced that S4C would broadcast the 2013 Final between Wrexham and Grimsby Town.

BT Sport showed the 2015 FA Trophy Final live. BT Sport also showed the 2016 FA Trophy Final between FC Halifax Town and Grimsby Town live on 22 May as part of a double-header along with the 2016 FA Vase Final.

title: FA Trophy


text: Category:Sulfates

title: Category:Sulfates


text:
Lotus Elan

Lotus Elan is the name of two separate ranges of automobiles produced by Lotus Cars. The first series of cars was produced between 1962 and 1975 as a rear-wheel drive vehicle. The second series was produced between 1989 and 1995 as a front-wheel drive vehicle.

Timeline
The first range of cars (1962–1975) comprised:
 * Two seater sports cars:
 * Lotus Type 26 drop head coupé (DHC) marketed as the Elan 1500, Elan 1600, and Elan S2 (Series 2).
 * Lotus Type 36 fixed head coupé (FHC) marketed as the Elan S3, the Elan S4 and, lastly, in a higher performance model, the Elan Sprint.
 * Lotus Type 45 drop head coupé, replacing the Type 26, delivered in parallel with the Type 36 in S3, S4 and Sprint form.
 * Lotus Type 26R racing version of the Type 26.
 * Four seater sports car (rear seats suitable for children):
 * Lotus Type 50, fixed head coupé, marketed as the Elan +2.

After the S2 was released the original Elan 1500 and Elan 1600 models were typically referred to as the S1 (Series 1) although the car was never explicitly marketed as such. Today, all models (S1-Sprint) are often cited collectively as the 1960s Elans.

The second range of cars (1989–1995) comprised:
 * Two seater sports cars:
 * Lotus Type M100 drop head coupé, initially marketed as the Elan S1 and, later, for the UK market, the Elan S2.

This second model was also produced in South Korea by Kia Motors between 1996 and 1999, rebadged as the Kia Elan.

Overview
The Lotus Elan was the first Lotus road car to use a steel backbone chassis with a fibreglass body. This style of construction was to be repeated in subsequent Lotus models for nearly three decades. At approximately 1500 lb, the Elan embodied Colin Chapman's minimum weight design philosophy. The Elan was technologically advanced with a DOHC 1,558 cc engine, four-wheel disc brakes, rack and pinion steering, and 4-wheel independent suspension. Gordon Murray, designer of the McLaren F1 supercar, reportedly said that his only disappointment with the McLaren F1 was that he could not give it the perfect steering of the Lotus Elan.

In 2004, Sports Car International named the Elan number six on the list of Top Sports Cars of the 1960s. The original version of the car was designed by Ron Hickman who also designed the first Lotus Europa as part of Lotus' GT40 project bid and made his fortune having designed the Black & Decker Workmate.

Because of its successful design and rigorous attention to cost control on the body, chassis, engine and transmission, the Elan become Lotus' first commercial success and contributed to the funding of its achievements in racing over the next ten years. It revived a company stretched thin by the more exotic, expensive to build, and rather unreliable Lotus Elite, which used a fiberglass monocoque body/chassis and all aluminium Coventry Climax engine.

The original Elan 1500 was introduced in 1962 as a roadster. After a very short production run of just 22 cars the engine was enlarged and the car was re-designated the Elan 1600. An optional hardtop was also offered. The Elan 1600 of 1963 was replaced by the Elan S2 in 1964. In 1965 the Type 36, a fixed head coupé version of the car, was introduced while in 1966 the drop head coupé Type 26 was replaced by the Type 45. Both Types, 36 & 45, were offered initially in S3 form, followed in 1968 in S4 form, and finally in 1970 as the Elan Sprint. Production of the Sprint ceased in 1973. The standard S2, S3 & S4 models were also available in a slightly more powerful and luxurious "Special Equipment" variant, generally referred to as the SE (e.g. Lotus Elan S3 SE).

In the UK the Elan was offered as a fully assembled vehicle and, for tax avoidance purposes, as a lower cost kit for final assembly by the customer.

Production
The total production number for the Lotus Elan is not definitively known; however John Bolster, in his book "The Lotus Elan and Europa: A Collector's Guide", provides a number of 12,224 (S1-3: 7,895; S4: 2,976; Sprint: 1,353). This number was occasionally used by Lotus itself. See below for +2 production.

Meanwhile, Paul Robinshaw and Christopher Ross, in their book "The Original 1962–1973 Lotus Elan", assert that Lotus' somewhat erratic record keeping at the time meant that vehicle serial numbers were not entirely sequential or consistent. Their assessment suggests the actual count to be in the range 8,676-9,153 (S1: 900; S2: 1,250; S3: 2,650; S4: 2976-3,000; Sprint: 900-1353).

As of April 2018, the voluntary, and thus inevitably incomplete, Lotus Elan registry lists approximately 1,100 known remaining vehicles (including approximately 330 +2 models) in over 30 countries.

Construction
The basic structure of the Lotus Elan comprised a fabricated mild steel backbone chassis, similar to a double ended tuning fork, and a fibreglass body. The chassis was the primary stressed component, providing the necessary bending and torsional rigidity. The fibreglass body was solidly bolted to the chassis at 16 points, fitting over it like a saddle. While not highly stressed the body nevertheless added to the overall rigidity of the structure. As such, the chassis should more properly be considered a subframe - it is readily changeable and most Elans on the road today have had a new chassis fitted at some point, either due to accident or decay.

The engine & gearbox are located between the front fork arms and the differential between the rear fork arms. The front & rear suspensions attach to the ends of their respective arms (turrets at the end of the arms hold the suspension springs & dampers).

This design resulted in light weight, high rigidity (by contemporary standards), and easy driver/passenger access through wide door openings with low sills. Driver and passenger protection from front and rear impact was acceptable for its time, but side impact protection was minimal.

Drive train
The engine, gearbox and differential were all sourced from Ford UK components. See below for details on the engine. The gearbox was a high volume unit used widely across the Ford range (Anglia, Cortina, etc.). Lotus offered regular and close ratio versions. The differential unit was also sourced from Ford but converted for independent rear suspension use (contemporary Ford cars used live rear axles). Three differential ratios were offered at various times during the production life of the vehicle: 3.55, 3.77 and 3.9. Most Elans were fitted with 3.77 differentials. 3.90 differentials, which provided the best acceleration, were often favored by purchasers of early cars, while 3.55 differentials became more common in later cars (especially the more powerful Sprints) to provide more relaxed cruising on newly built highways.

The final drive used four Rotoflex couplings to connect the differential output shafts to the rear hubs. These "rubber doughnut" couplings were widely used at the time for road vehicles (e.g. Hillman Imp, Triumph GT6) and racing (e.g. Ford GT40, Lotus 21), prior to the availability of constant velocity (CV) joints. In the case of the Elan, which had exceptionally supple rear suspension that allowed significant vertical wheel travel, the deformation of the Rotoflexes in operation resulted in some "wind up" of the couplings. This could be readily detected by the driver, especially at takeoff and during gear changes, and was slightly disconcerting when driving an Elan for the first time. In practice drivers typically adjusted their clutch technique within a few minutes and no longer noticed it. In recent years the uncertain quality of replacement Rotoflex couplings, combined with the availability of half-shafts built with CV joints, has resulted in many Elans being converted from Rotoflex to CV joint drive.

A notable feature of the drive train design was its use of standard, mass-produced components in combination with a minimal number of specialized aluminium castings to create a power unit and transmission suitable for a high performance sports car:
 * 1) The engine head and timing chest. Converted the Ford Kent pushrod engine into a twin overhead cam engine.
 * 2) The differential rear cover & carrier. Converted a live axle to independent suspension operation.
 * 3) The rear wheel hubs. Required for the Chapman Strut rear suspension.

These three items were unique to the Elan (although the engine was subsequently used widely in other vehicles).

Engine
The 1,558 cc "Lotus TwinCam" engine was based on the Ford Kent Pre-Crossflow four-cylinder 1,498 cc engine, with a Harry Mundy-designed two-valve alloy chain-driven twin-cam head. The rights to this design was later purchased by Ford, which renamed it the "Lotus-Ford Twin Cam". It would go on to be used in a number of Ford and Lotus production and racing models.

Lotus reported different power outputs for the Twin Cam engine during the production of the Elan. Prior to the release of the Sprint the following outputs were reported in the Workshop Manual:

Lotus marketing material from the S3 period quoted the SE variant at 115 bhp, noting high lift cams, carburetor re-jetting and a four branch exhaust.

The final version of the Workshop Manual revised these numbers, as shown in the following tables. The increased power of the Sprint engine was primarily due to slightly larger inlet valves, so it was named the Big Valve engine (it has been widely reported that the combination of 'hotter cams', rejetting the carbs and big valves improved performance of the 115 bhp SE by only 11 bhp. As a marketing exercise to increase sales of the new Sprint model, Chapman saw to it that published bhp figures for existing models be downgraded to increase the apparent improvement in performance):

Suspension, steering, braking
The Elan utilized modern technologies for its suspension, steering and brakes. The overall design was simple and easily maintained. In common with many Lotus cars they were perhaps the Elan's most highly regarded features.
 * Front suspension: The front suspension was based on Triumph wheel uprights & steering components while the remaining pieces were of Lotus design. The layout was a classic double wishbone arrangement with coilover springs and shock absorbers. Each "wishbone" consisted of two separate arms, with the upper pair connected to a ball joint at the top of the suspension upright and the bottom pair connected to a trunnion joint at the base of the upright. The bottom pair also held the base of the shock absorber and coil spring assembly (the top being connected to its chassis suspension tower). Similarly, the steering was of classic rack and pinion design, the Triumph rack (originally acquired from Alford & Alder) being modified for the Elan's narrow track with custom track rods and rack collars that set the minimum turning radius. The steering, 2 $2/3$ turns lock-to-lock, was not power assisted, which allowed for the visceral steering feel for which the Elan was famous.
 * Rear suspension: The Chapman Strut rear suspension was designed and manufactured by Lotus. A single brazed "A frame" wishbone was bolted to each side of the base of a cast aluminum upright/wheel-hub, and to the chassis at two, widely set, points. The shock absorber and coil spring strut was integrated into the upright and bolted via a rubber "Lotocone" coupling to its chassis suspension tower. This resulted in a simple design with just three chassis connection points.
 * Brakes: The disc brakes (9.5 in front and 10.0 in rear) were supplied by Girling. Most Elans used a single hydraulic circuit although Federal cars were fitted with dual circuits. Early Elan brake systems were not servo assisted, but servos were fitted to SE and later models. In practice the Elan was light enough that power assistance for the brakes was not required, so the provision of servo assistance may have offered a marketing benefit as much as a functional benefit.

Major component suppliers
Lotus relied heavily on suppliers of mass-produced parts to create the Elan. These included:
 * Engine/Gearbox/Differential from Ford UK.
 * Front suspension and steering from Triumph (Herald/Spitfire/Vitesse).
 * Electrical systems from Lucas (dynamo, starter, wiring, relays, ignition, lighting, etc.).
 * Instrumentation & other electrical from Smiths (speedometer, tachometer, heater, etc.).
 * Carburetors from Weber, Dell'Orto and Zenith-Stromberg (depending on model).
 * Brakes from Girling.

Weight
As with all Lotus cars, attention to minimizing weight was a major factor during the Elan's design and manufacture. Its diminutive size (while offering room for drivers over 6 ft tall and practical luggage carrying capacity) provided the foundation for its low weight.

The Workshop Manual lists the following weights:

Lotus marketing material from the S1 period quoted the weight as "11 $1/2$ cwt" (1288 lb) and from the S3 period quoted as "from 1,232 lb" (558 kg).

Meanwhile, reported weights in magazine road tests varied from 1260 lb (Elan 1600, Cars Illustrated, August 1964) to 1630 lb (S4 FHC SE, Road & Track, January 1969).

The figures above show that the weight of the vehicle changed during its production lifetime, in general tending to increase. Additional features such as electric windows, more luxurious carpeting, and larger wheels and tyres all contributing to the gain.

Sub assemblies

 * Chassis: 88 lb - anecdotal reports in www.lotuselan.net suggest that the weight was around 75 lb.
 * Body: 199 lb (per contract with S. Bourne & Co for early body shell construction, with subsequent variations due to DHC/FHC and S1-4 model differences).
 * Door shell (without window assembly): 11 lb
 * Hood/bonnet: 13.5 lb
 * Boot/trunk lid: 12 lb
 * Engine: 260 lb.
 * Gearbox: 84 lb.
 * Differential: 46 lb.

Performance
The performance of the Elan was derived from the combination of its powerful engine (by contemporary standards) and light weight. The roadholding and agility of the car also meant that high speeds could be maintained on corners, which allowed for high average speeds. By contemporary standards, the Sprint was an exceptionally rapid car and quite lived up to its name. It embarrassed almost every other supercar at the time in terms of outright acceleration up to about 90 mph. This electrifying performance was accentuated by the brilliant handling and road holding, allied to the small size and weight of the Elan, which meant that a well driven Elan on a dry road could outclass any other non-Lotus road car.

Although performance results achieved by testers are affected by many variables (e.g. differential ratio, weather/road conditions, gross vehicle weight, DHC vs. FHC, etc.) the following tables provide an overview of the car's capabilities.

The following excerpt from Robinshaw & Ross summarizes Elan performance for each Series:

A few additional figures for the Sprint:

Reviews
The Elan was widely admired and praised by customers and reviewers, noted for its exceptional handling, roadholding, steering, acceleration, braking and comfort: Car and Driver: The Elan very simply represents the sports car developed in tune with the state of the art. It comes closer than anything else on the market to providing a Formula car for ordinary street use. And it fits like a Sprite, goes like a Corvette, and handles like a Formula Junior. Driving it is very simply another sort of automotive experience altogether. Most people tend to come back from their first ride a little bit glassy-eyed... Road and Track: The light and tactile steering, combined with supple suspension and a weird, physics-defying sense of zero weight transfer in corners, provides a sensation akin to flying just over the ground. I'm convinced there's a powerful pleasure center in the brain that remains untapped until you drive an Elan. It's almost a drug. Motor Sport: The tremendously responsive steering and handling requires similar qualities from the driver and the speeds achieved round corners and on the straight are deceptively fast. This, therefore, calls for a lot of concentration on the driver's part. Once mastered, however, the Elan is the nearest thing to a single-seater racing car one is likely to be able to drive comfortably on the road. To master the car and explore its tremendous handling potential along that delightfully twisty piece of road one knows so well is close on perfection for the sporting motorist.

Influence
While the structure of the Elan followed an entirely traditional approach for sports cars of the time - front engine, rear wheel drive - its design included novel ideas that found their way into the designs of other manufacturers' vehicles.

Examples include:
 * The Mazda MX-5 (Mazda Miata in North America). The original Elan is usually credited as being the design inspiration for this sports car in 1989. Two Elans were intimately evaluated by Mazda in the process of designing the MX-5.
 * The Toyota 2000GT. This sports car used a chassis that bears a striking resemblance to the Lotus Elan.

The car designer and engineer Gordon Murray said "Series 3 Lotus Elan... it's still, in my opinion, probably the best-handling sports car that's ever been made...  If anybody wants to know what good steering is, just jump in the 60s Elan."

Owners, drivers and in the media
This generation of the two-seater Elan was famously driven by Diana Rigg in the character of Emma Peel in the 1960s British television series The Avengers.

The reference to a car accident in the Beatles song "A Day in the Life" was apocryphally based on Tara Browne's fatal accident in his Lotus Elan.

Famous celebrities, past & present, who owned and/or drove a Lotus Elan include:
 * Peter Sellers - English comedian.
 * Jim Clark - Scottish racing driver.
 * Paul Newman - American actor.
 * Jay Leno - American TV personality.
 * Michael Crawford - English actor, singer, comedian.
 * Noel Redding - English rock musician (Jimi Hendrix Experience & Fat Mattress).

Racing
Despite the fact that the Lotus Elan has been (and continues to be ) used extensively for racing it was Lotus' first car that was not designed with racing in mind. (The earlier Lotus Elite was designed as a road car and also to compete in high-efficiency classes at Le Mans. ) Nevertheless, because owners assumed that all Lotus cars were designed for racing, it soon found its way onto the track, however unsuitable. Lotus resisted modifying the car to make it more suitable for racing but eventually created a racing version of the Elan. Robinshaw/Ross quote Colin Chapman:"'When we announced the Elan we said 'This is a touring car, it is not intended for racing and have done no competition development on it.' The fact that customers bought them and tried to race them was originally no concern of ours, but in the second year we thought, well, if these people insist on racing them then we'd better get down to some proper development. They were too softly sprung, too softly damped, tore their doughnuts apart and had all sorts of drama, but it was the name, and people thought they must be racing cars.'"

The result of the "proper development" was the Type 26R version of the Elan, offered from 1964 at £1995 in kit form. "Motor Sport: Mechanically, the 26R differed by featuring racing lightweight competition-spec wishbones, sliding spline driveshafts in place of rubber joints, bigger anti-roll bars and a degree of reinforcement around the suspension pick-up points. Pedals were repositioned to aid heel-and-toeing, dual circuit brakes with twin master cylinders and light alloy calipers coming as standard. As did a 140bhp Cosworth-tuned 'four' although up to 160bhp was offered in time."

Other changes included flared wheel arches, which allowed for larger wheels and tires, and a lighter body shell.

Series (model) differences
The table below provides a simplified summary of Elan updates and changes associated with each Series. It is notable that the fundamentals of the car changed very little during its eleven years of production.

Each Series provided a general refresh of the car, incorporating new, modern features (e.g. electric windows, larger tires, power washers & windows, hazard & reversing lamps) and updated cosmetics (e.g. improved sound proofing and carpeting, improved DHC hood (roof) design, Sprint color scheme).

However, as a low volume manufacturer, Lotus' processes were flexible enough to change the components used during construction at will. As a result, significant variations could exist between cars of the same Series.

Ongoing changes were frequently necessary for a variety of reasons: component availability from suppliers; price changes; the need to meet regulatory requirements (especially for Federal vehicles); model year enhancements; and the phase-in & phase-out of stock during the transition to the next Series.

Lotus promoted the potential variation between vehicles by stating in its documentation: Lotus policy is one of continuous product improvement and the right is reserved to alter specifications at any time without prior notice.

Price
The price of the Elan changed during its production lifetime. At announcement, in October 1962, the fully built cost was £1,499, although between 1963 and 1966 the factory price was generally 5-10% lower. By the end of production, in April 1973, the fully built cost had increased to £2,436.

However, throughout its production the Elan was also available in the UK in "kit" form. This was provided for tax avoidance reasons and required the customer to complete the final assembly of the vehicle. The tax savings were considerable, the prices above being reduced to £1,095 and £2,044 respectively.

Kit versions of the Elan were as complete as possible while adhering to tax regulations. The entire chassis, body, electrical systems, final drive/suspension and all trim were delivered fully assembled to the customer. Final construction involved installation of:
 * Engine, gearbox & prop shaft.
 * Front wheel hubs, springs/shock absorbers & anti-roll bar.
 * Exhaust.
 * Wheels/tyres.
 * Battery.
 * Radiator & hoses.

For the typical Lotus customer, an auto enthusiast, final assembly would be undertaken willingly. When complete the owner would take the car to a Lotus dealer who, after checking that the assembly had been performed correctly, initiated the warranty period. This feature of Elan purchase was quite heavily promoted by Lotus, with print advertisements showing how simple and rewarding it was to complete the construction.

Comparative pricing
'The Motor' magazine regularly printed prices for all UK cars. A partial extract from its pages provides comparative pricing for several sports cars of the Elan S1 period. The table also shows that the Elite was significantly more expensive than the Elan:

Collectibility
1960s/70s Elans are now passing half a century old and, as a well respected Lotus and a fine example of Colin Chapman's design philosophy, are valued today as collectors' vehicles. In the UK they have an almost cult-like status among auto enthusiasts, commanding higher prices than elsewhere in the world. An undocumented number of right hand drive (UK market) Elans have been privately exported to Japan, where they are admired by collectors. As with many English sports cars of the period there are dedicated enthusiasts around the world, especially in Australia, Canada and the USA.

The simple design of the car, with nearly all parts being readily replaceable, makes it well suited for the collector market. New chassis, bodies and drive train components are all available. There are numerous suppliers of Elan parts and restoration services, worldwide. Conveniently, the fiberglass body is immune to rust.

Current Values
Current values vary widely around the world, so the following table provides approximate figures for the UK, based on condition. Early cars (S1 & S2) and Sprints are more highly valued than S3s, which are more highly valued than S4s.

Generally, Drop Head Coupés have been more highly valued than Fixed Head Coupés and in the past it was not uncommon to convert FHCs to DHCs, making FHCs rarer. This has tended to stabilize the value of FHC cars.

On 25 February 2023, Silverstone Auctions Ltd. auctioned seven fully restored Lotus Elans, each featuring a well regarded provenance, from the Piddington Collection. The prices fetched, before 12.5% buyers premium and VAT were:

 * 1975 Sprint FHC last ever Elan - £40,000
 * 1971 Sprint DHC ex-Ron Hickman - £58,000
 * 1969 S4 FHC ex-Jochen Rindt - £58,000
 * 1966 S3 DHC ex-Diana Rigg (Emma Peel/Avengers) - £146,000
 * 1968 S3 DHC ex-Rob Walker - £44,000
 * 1968 S4 FDC ex-Keith Duckworth - £32,000
 * 1966 S3 DHC ex-Peter Sellers - £66,000

Gallery
Elan 1600 and S2:

Overview
An Elan +2 was introduced in 1967 with a longer wheelbase and wider track, and two small rear seats, suitable for children. While the Elan was a two-seater sports car marketed to the single man/woman or young couple, the +2 was larger and considerably more luxurious vehicle marketed to the established family with children. Reviews reflected this new marketplace for Lotus. Motor Sport magazine reported the design goal as: it was decided to increase the interior dimensions considerably to meet the car's maxim that it "must be capable of transporting two adults and two children 1,000 miles in comfort with their luggage". In 1971 Colin Chapman bought Moonraker Marine / JCL and in 1974 Brian Davey working with the Naval Architect Don Shead on a new boat design was asked to help redesign the replacement for the Europa. This was finally designed by Giugiaro of Italdesign and called the Esprit which started production in 1976.

While the Elan ceased production in 1973 the Elan +2 continued to be produced until 1975.

Construction
Aside from the increase in size necessary to accommodate two additional seats, the design of the Elan +2 was extremely similar to the two seater Elan. The concept of a backbone chassis with fibreglass body, powered by a Ford UK-derived drive train was maintained. Double wishbone and Chapman Strut front/rear suspension of identical design was also used, with some components (e.g. rear hubs) shared with the two seater Elan and others (e.g. wishbones) increased in size. Power assisted brakes were standard, dual circuit in some markets.

In 1968 a more luxurious version of the Elan +2 was released, named the +2S.

In 1971 +2S was upgraded to include the Big Valve engine, and then named the +2S 130. Later models of the +2S 130 were provided with a 5-speed Austin Maxi based gearbox, which greatly improved its high speed cruising capabilities, named +2S 130/5.

Performance

 * Tested maximum power: 108-126 bhp net (depending on the model).
 * Top speed: 120 mi/h.
 * Acceleration: 0-60 mi/h in 7.9 seconds, 0-100 mi/h in 21.8 seconds.

Production
As with the two seater Elan, the exact number of +2s produced is unknown. The estimated total is approximately 5,200. However, John Bolster's "The Lotus Elan and Europa: A Collector's Guide" provides a number of 3,300. Fewer than 1,200 of these cars remain on the roads today.

Elan (M100)
The Lotus M100 series Elan was launched in August 1989, reviving the Elan nameplate after 14 years. A two-seater convertible sports car with front-wheel drive, designed in-house by Lotus, it featured an engine and manual transmission supplied by Isuzu, and was built with the development and testing resources of General Motors. Around £35 million (about $55 million) was invested in its development, more than any other car in Lotus history. Its design, featuring a fibreglass composite body over a rigid steel backbone chassis, was true to Lotus founder Colin Chapman's original philosophy of achieving performance through low weight, and the name "Elan" connected the car with its 1960s ancestor.

Origins
In 1986 the purchase of Lotus by General Motors provided the financial backing to develop a new, small, affordable car in the same spirit as the original Elan (last built in December 1972). A development prototype, the M90 (later renamed the X100) had been built a few years earlier, using a fibreglass body designed by Oliver Winterbottom and a Toyota-supplied 1.6-litre engine and transmission. The M90 was in turn preceded by an earlier two-seater project in 1974 called the Lotus M80, also designed by Winterbottom and based on the Lotus Eclat. However money was in short-supply at Lotus and development was soon stopped, with the M80 name later resurfacing on the 1984 Lotus Eminence M80 four-door V8 saloon concept. Lotus was hoping to sell the car through Toyota dealerships worldwide, badged as a Lotus Toyota, but the project never came to fruition and the prototype was shelved (although Lotus' collaboration with Toyota had some influence on the design of the Toyota MR2).

The idea of a small roadster powered by an outsourced engine remained, however, and in late 1986 Peter Stevens's design for the Type M100 was approved and work began by Lotus engineers to turn the clay styling buck into a car that could be built. This process was completed in just under three years, a remarkably short time from design to production car.

Testing
The new Elan was conceived as a mass-market car and in particular one that would appeal to US buyers. Consequently, Lotus put an enormous effort (for such a small firm) into testing the car; over a two-year period 19 crash cars and 42 development vehicles were built, logging nearly a million test miles in locations from Arizona to the Arctic. The Elan was driven at racing speeds for 24 hours around the track at Snetterton. Finally each new car was test-driven for around 30 mi at Lotus' Hethel factory to check for any manufacturing defects before being shipped to dealers.

Handling
The choice of front-wheel drive is unusual for a sports car, but according to Lotus sales literature, "for a given vehicle weight, power and tyre size, a front wheel drive car was always faster over a given section of road. There were definite advantages in traction and controllability, and drawbacks such as torque steer, bump steer and steering kickback were not insurmountable." This was the only front-wheel-drive vehicle made by Lotus. Every model made since the M100 Elan, such as the Lotus Elise, has been rear-wheel drive.

The M100 Elan's cornering performance was undeniable (on release the Elan was described by Autocar magazine as "the quickest point to point car available"). Press reaction was not uniformly positive, as some reviewers found the handling too secure and predictable compared to a rear-wheel-drive car. However, the Elan's rigid chassis minimised roll through the corners and has led to many critics describing it as 'the finest front wheel drive [car] bar none'. Unlike the naturally aspirated version, the turbocharged SE received power steering as standard, as well as tyres with a higher ZR speed rating.

Engine
The M100 Elan used a 1,588 cc double overhead camshaft (DOHC) I4 16-valve engine, sourced from the Isuzu Gemini and extensively modified by Lotus (a third generation of this engine was later used in the Isuzu Impulse), which produced 162 hp in turbocharged form. 0–60 mph acceleration time was measured by Autocar and Motor magazine at 6.5 seconds, and a top speed of 137 mi/h was recorded.

Significant differences in the Isuzu-Lotus engine from the original include a new exhaust system, re-routed intake plumbing for better thermodynamic efficiency, improved engine suspension, and major modifications to the engine control unit to improve torque and boost response. Almost all models featured an IHI turbocharger.

Braking and suspension components
The majority of the suspension items for the M100 Elan (both S1 and S2) were manufactured and supplied by Portland Engineering based in Dorset in the UK. They supplied front and rear brake discs, rear suspension arms, front and rear hub carriers along with numerous other components. The rear brakes had an unusual design in that the rear hub was incorporated within the brake disc.

Sales
Two variants were available at launch, the 130 bhp Elan 1.6 (retailing at £17,850) and the 162 bhp Turbo SE (£19,850). Initial sales were disappointing, due to the debut of the more affordable "nostalgic" Mazda MX-5 which was arguably similar in concept to the 1960s Elan, in contrast to the M100's deliberately futuristic cant. The Elan was very expensive to make (the cost to design and produce the dashboard alone was more than the total cost of the Excel production line), and sales figures were too low to recoup its huge development costs.

Altogether just 3,855 Elans were built between November 1989 and July 1992, including 129 normally aspirated (non-turbo) cars. A mere 559 of them were sold in the US, featuring a 'stage 2 body' which had a different rear boot spoiler arrangement together with a lengthened nose to accommodate a USA-compliant crash structure and airbag, and 16-inch wheels (optional in most markets, standard in the U.S.) instead of 15-inch as on the UK model. A coupé version of the Elan, to replace the Excel, had also been in the pipeline. This, too, was shelved, and there was no direct successor to the Excel, which ended up also being discontinued in 1992.

Series 2
Two years after the end of the original production run, a limited edition of 800 Series 2 (S2) M100 Elans was released during the Romano Artioli era (produced June 1994–September 1995) when it was discovered that enough surplus engines were available to make this possible. It was only for the UK market. According to Autocar magazine, the S2 addressed some of the concerns over handling, but power was reduced to 155 bhp and the 0–60 mph acceleration time increased to 7.5 seconds, due to the legislative requirement to fit a catalytic converter in all markets. The S2s have very similar performance to the USA vehicles, having an identical engine management system calibration albeit with a slightly lower overall vehicle weight.

Kia Elan
After the final production run of the Elan in 1995, Lotus sold its production rights to Kia Motors, which produced its own version. Outwardly, the Kia Elan looks almost identical to the original. The most obvious difference are the Kia-designed taillights which replaced the Renault Alpine GTA rear lights of the original.

Kia Motech (Kia Motor-technology) produced the car in Ansan, South Korea from 1996 to 1999 as the Kia Elan for the Korean market, using a 151 PS 1.8 L T8D engine of Mazda origins instead of the Isuzu 1.6 turbo-charged unit. In the Japanese market, the car was sold as the Vigato. A total of 1,056 were produced.

2013 Elan concept show car
A new Lotus Elan was announced at the 2010 Paris Motor Show. It was hoped to be in production by 2013, but it, alongside the Esprit and Elite concepts, was cancelled before it could enter production. The car was to have featured a 4.0-litre V6 engine with around 350 horsepower, a 0-60 time of 4.5 seconds, a top speed of 165 mph, and was to have weighed roughly 1295 kg. The car was also meant to have a new kinetic energy recovery system (KERS) which would provide short bursts of extra passing power using energy recovered during braking.

title: Lotus Elan