Transformations Lorentz

We consider two inertial frames of reference, S and S, with origins O and O and axes Ox, Ov, Oz in S and Ox. OV, O in S. (An inertial frame of reference is defined as a coordinate frame in which the laws of Newtonian mechanics hold one of the consequences of the special theory of relativity is that any pair of such inertial frames can only move with a uniform velocity relative to each other.) Now an observer at the origin O will describe an event in his frame by values of x, y, z, t where t is the time measured by a clock at rest in S. Similarly an observer at O will describe the same event in terms of the corresponding values x, y, z, t measured in S.  

We are interested in the relationship between observations in S and S when the two frames of reference are in uniform relative motion. For the sake of simplicity, we 

In other words, it is the distance, ct, travelled by light in a given time interval which fulfills the role of the fourth coordinate, rather than the time interval itself. The special theory of relativity requires that after a Lorentz transformation the new form of all laws of physics is the same as the old form. The Dirac equation, for example, is invariant under a Lorentz transformation. 

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Before outlining Toffoli s model of a deterministic relativistic diffusion C A model, we motivate the discussion by recalling a simple formal analogy that holds between a circular rotation by an angle 6 in the x,y) plane and a Lorentz transformation with velocity... 

It relates the space time coordinates xf of an event as labeled by an observer 0, to the space-time coordinates of the same event as labeled by an observer O . The most general homogeneous Lorentz transformation is the real linear transformation (9-8) which leaves invariant the quadratic form... 

These subsets are disjoints and cannot be continuously connected by real Lorentz transformations. 

The set of all inhomogeneous Lorentz transformations form a ten-parameter group, usually called the Poincar6 group. 

Note that the scalar product is formally the same as in the nonrela-tivistic case it is, however, now required to be invariant under all orthochronous inhomogeneous Lorentz transformations. The requirement of invariance under orthochronous inhomogeneous Lorentz transformations stems of course from the homogeneity and isotropy of space-time, send corresponds to the assertion that all origins and orientation of the four-dimensional space time manifold are fully equivalent for the description of physical phenomena. 

To make these notions precise, the transformation properties of the wavefunction x under spatial and time translations as well as under spatial rotations and pure Lorentz transformations must be specified and it must be shown that the generators of these transformations form a unitary representation of the group of translations and proper Lorentz transformations. This can in fact be shown5 but will not be here. 

We next inquire as to the transformation property of the quantity ip(z) under a homogeneous Lorentz transformation... 

Now one readily verifies that for proper Lorentz transformations the matrices... 

Equation (9-369) allows us to infer that the transformation of the adjoint spinor under Lorentz transformation is given by... 

Hence, under a proper homogeneous Lorentz transformation -without time inversion the quantity transforms like a scalar ... 

The T corresponding to various infinitesimal transformations (e.g., an infinitesimal rotation about the 2-axis, or an infinitesimal Lorentz transformation about the x-axis) can be explicitly computed from this representation. The finite transformations can then be obtained by exponentiation. For example, for a pure rotation about the 1-direction (x-axis) through the angle 6,8 is given by... 

For a pure Lorentz transformation along the 1-direction corresponding to a hyperbolic angle 6,8 has the form... 

It should be noted that 8 is unitary and two-valued for a pure rotation, and is hermitian (but single-valued and nonunitary) for a pure Lorentz transformation. 

Wightman, A. S., and Sohweber, S. S., Phye. Bee., 98,812 (1954). A discussion of the transformation properties of the operators and under Lorentz transformation is also included in this reference. 

For a discussion of the transformation of the field operators under improper Lorentz transformations and discrete symmetry operations such as charge conjugation, see ... 

The formalism can be carried farther to discuss the particle observables and also the transformation properties of the s and of the scalar product under Lorentz transformations. Since in our subsequent discussion we shall be primarily interested in the covariant amplitudes describing the photon, we shall not here carry out these considerations. We only mention that a position operator q having the properties that ... 

Consider next the relativistic invariance of quantum electrodynamics. Again, loosely speaking, we say that quantum electrodynamics is relativistically invariant if its observable consequences are the same in all frames connected by an inhomogeneous Lorentz transformation a,A ... 

It can actually be shown that the factor to can be chosen to be l.s Furthermore, for restricted Lorentz transformations Via, A) must be unitary since every element of the group a, A can be written as the product of elements that are the same... 

The previous results become somewhat more transparent when consideration is given to the manner in which matrix elements transform under Lorentz transformations. The matrix elements are c numbers and express the results of measurements. Since relativistic invariance is a statement concerning the observable consequences of the theory, it is perhaps more natural to state the requirements of invariance as a requirement that matrix elements transform properly. If Au(x) is a vector field, call... 

The (four-vector). au(x) represents the result of some local measurement at the point x performed by a (Lorentz) observer 0. An observer O (related to 0 by a Lorentz transformation x = Ax) describes this measurement by... 

We have noted that the unitary operators U(a,A) define a representation of the inhomogeneous group. If we denote by P and AT, the (hermitian) generators for infinitesimal translations and Lorentz transformations respectively, then... 

The right side is again determined by the fact that tjr(x) must transform like a spinor under a homogeneous Lorentz transformation. Its form is made transparent by recalling that for an infinitesimal homogeneous Louentz transformation... 

Similar considerations lead to the transformation properties of the one-photon states and of the photon in -operators which create photons of definite momentum and helicity. We shall, however, omit them here. Suffice it to remark that the above transformation properties imply that the interaction hamiltonian density Jf mAz) = transforms like a scalar under restricted inhomogeneous Lorentz transformation... 

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