Lorentz-invariant scaling hypothesis and some implications

Abstract

In this article we show that one can formulate a Lorentz-invariant scaling hypothesis which is consistent with that of Yang and Feynman. When this is applied to some existing single-particle distribution functions, one concludes that parameters in these distributions-such as "temperature"-should be Lorentz scalars, and therefore d o not necessarily have the physical attributes usually given to them.T h e purpose of this article is to show that one can formulate a Lorentz-invariant scaling hypothesis and that, as a result o f it, the parameters entering into some single-particle distributions, such as "temperature," should be Lorentz scalars and therefore do not necessarily have the physical attributes usually given to them.As is well known, the single-particle inclusive reaction can b e most simply described by the Lorentzinvariant single-particle inclusive distribution function where Tk (the momentum) and d (the energy) r e f e r specifically to particle c. Despite the fact that p i s a Lorentz-invariant quantity, the usual scheme of the analysis directly through the Mandelstam invariants appears to b e rather complicated, f o r , in general, it requires three independent variables.' However, specifying the Lorentz f r a m e s to be either the laboratory s y s t e m (Benecke el a L 2 ) or the c.m. (center o f m a s s ) s y s t e m o f particles (1 and b (Feynman3), it was shown that one can e ffectively analyze p in t e r m s of only two variables. The v e r y fact that p can be successfully analyzed in t e r m s o f only two variables allows one to construct rather easily model-based explicit f o r m s distribution5 i s a true temperature, by including in the discuss ion the "quantum-mechanical'' distribution o f Takibaev el u L .~ we demonstrate that one can "deduce" distribution f r o m rather div e r s e models. For this reason alone one should not take too seriously the given physical attributes of various parameters entering into these distributions (of course, we intend to show this by invoking the Lorentz-invariant scaling hypothesis). Let us f i r s t describe brieflv the quantum-mechanical distribution p of Takibaev el ~1 .~ They assume that particle c may be described by the minimized wave packet in variable I;, which e xactly satisfies the uncertainty equation ( i n the c.m. s y s t e m o f the two colliding particles (1 and 6). After taking into account that particles (1 and h (say, nucleons) are shortened by the contraction factor J G 2 m ( s i s the square o f the c.m. energy and I?L is the m a s s o f the nucleon, either a or 6 ) along the longitudinal axis prior t o the collision, Takibaev r l c 1 L 4 arrive at the expression (valid in the c.m. s y s t e m o f nucleons) Ln ( 3 ) A i s some normalization constant, 1 2 , i s the transverse momentum of particle c with respect to the line of the collision, ( k . 2 ) is "the mean square value of the transverse momentum," and f o r p, mostly in the c.m. system. Here we wish is the Feynman variable, where k t i s the comt o present two rather diverse f...

The proper choice of the Lagrangian for a relativistic particle in external fields

“…In a very readable discussion, Logan (1977) shows that the necessary and sufficient condition for a Lagrangian to be parameter invariant is that it be homogeneous of degree + 1 in X' ,,. By Euler's theorem on homogeneous functions, this in turn implies the identity l ?…”

“…Rather, we would employ the requirement of parameter independence and, therefore, the requirement of homogeneity of degree + 1 in X'" in order to restrict the possible forms of the desired Lagrangian (Bergmann 1962, Rzewuski 1964, Rohrlich 1965. Sudarshan and Mukunda 1974, Logan 1977. As a second fundamental principle to guide us in the choice of a suitable Lagrangian, we must of course comply with the principle of relativity, which tells us that the equations of motion are to preserve their form when subjected to a Lorentz transformation.…”