Klein-Gordon

In going from the Schrodinger equation to the Klein-Gordon equation, we obtain the neeessary symmetry between spaee and time by having seeond-order derivatives throughout. It is usually written in a form that brings out its relativistic invarianee by using what is ealled/our-vector notation. We define a four-vector X to have components... 

Don t confuse this with my earlier use of x for a space-spin variable the notation is common usage in both applications.) The Klein-Gordon equation is therefore... 

It turns out that the Klein-Gordon equation cannot describe electron spin in the limit of small kinetic energy, it can be shown to reduce to the familiar Schrodinger equation. 

The Schrodinger equation and the Klein-Gordon equation both involve second order partial derivatives, and to recover such an equation from the Dirac equation we can operate on equation 18.12 with the operator... 

A little operator algebra shows that this gives exactly the Klein-Gordon equation if the y s satisfy the relationship... 

It should be stressed that the amplitudes x introduced above are not the usual covariant amplitudes. Their relation to the more familiar covariant Klein-Gordon amplitude for a spin 0 particle and the covariant Dirac amplitude for a spin particle will be discussed at the appropriate place. In the next section we turn to a discussion of the covariant amplitudes describing spin 0 particles. 

Spin 0 Particles.—The covariant wave equation describing a spin 0, mass m particle is the Klein-Gordon equation ... 

Although the Klein-Gordon equation is of second order in the time derivative, for a positive energy particle the knowledge of at some given time is sufficient to determine the subsequent evolution of the particle since 8ldt is then given by Eq. (9-85). Alternatively Eq. (9-85) can be adopted as the equation of motion for a free spin zero particle of mass m. We shall do so here. 

This scalar product is conserved in time if and 2 obey the Klein-Gordon equation. It furthermore possesses all the properties usually required of a scalar product, namely... 

However, in order to give an unambiguous answer to the question of how one is to calculate the probability of finding a Klein-Gordon particle at some point x at time t, we must first find a hermitian operator that can properly be called a position operator, and secondly find its eigenfunctions. It is somewhat easier to determine the latter since these should correspond to states wherein the particle is localized at a given point in space at a given time. Now the natural requirements to impose on localized states are ... 

Let < j(k) be the Klein-Gordon amplitude corresponding to a spin zero particle localized at the origin at time t = 0. Since in momentum space the space displacement operator is multiplication by exp (— tk a), the state localized at y at time t = 0 is given by exp (—ik-y) (k). This displaced state by condition (b) above must be orthogonal to (k), i.e. 

Since in (9-150), k2 = m2, i > x) also satisfies the Klein-Gordon equation... 

Klein-Gordon amplitude relation to Schrodinger amplitude, 500 Klein-Gordon equation for destruction operator, 507... 

Schrodinger amplitude relation to Klein-Gordon amplitude, 500 Schrodinger equation, 439 adiabatic solutions, 414 as a unitary transformation, 481 for relativistic spin % particle, 538 for the component a, 410 in Fock representation, 459 in the q representation, 492 Schrodinger form of one-photon equation, 548... 

This procedure leads to the Klein-Gordon equation... 

In the case of the free scalar field, since we have equation of motion for the tilde and non-tilde variables, the f3—dependent Klein-Gordon field theory is given by the Lagrangian (Y. Takahashi et.al., 1975 H. Umezawa, 1993)... 

The formalism can be extended for a quantum Jield with the TFD Lagrangian density given by t = — , where is a replica of for the tilde fields so leading to similar equations of motion. For the purpose of our applications, we shall restrict our analysis to free massless fields. Thus, considering the free-massless boson (Klein-Gordon) field, the two-point Green function in the doubled space is given by... 

The approach presented in this work can be also extended to the case of Dirac and Klein-Gordon operators, too in order to calculate finite-temperature spectra of heavy-light and hybrid mesons. 

The Dirac equation is invariant to Lorentz transformations [8], a necessary requirement of a relativistic equation. In the limit of large quantum numbers the Dirac equation reduces to the Klein-Gordon equation [9,10]. The time-independent form of Dirac s Hamiltonian is given by... 

The spin-independent part of these equations is identical to the Klein-Gordon equation. If the singularity of V is not stronger than 1/r then,... 

The magnetic fluxes F and G obey the Klein-Gordon equation for a massless particle in the vacuum ... 

In this section we describe the general approach to constructing conformally invariant ansatzes applicable to any (linear or nonlinear) system of partial differential equations, on whose solution set a linear covariant representation of the conformal group 0(1,3) is realized. Since the majority of the equations of the relativistic physics, including the Klein-Gordon-Fock, Maxwell, massless Dirac, and Yang-Mills equations, respect this requirement, they can be handled within the framework of this approach. 

Incidentally, the telegrapher s equation (17) with x = ihjlmc2 is satisfied by the Klein-Gordon (also Dirac) wavefunction for a free particle, if the factor exp [—(ijh)mc2f is split off from it. Thus, the time lag according to relativity corresponds to an imaginary relaxation time x. 

---