Oct 09, 2016 the lagrangian formalism is one of the main tools of the description of the dynamics of a vast variety of physical systems including systems with finite particles and infinite number of degrees of freedom strings, membranes, fields. It is the elucidation of the interplay between these, along with generalizing previous results, that is the present goal. Anyway, what they had to do without mentioning it is to integrate by parts equiv. The scheme is lagrangian and hamiltonian mechanics. That would lead us back to the problem of negative energy states. To obtain the dynamical equations from a lagrangian density we use hamiltons principle of least action. Dirac equation dirac placed emphasis on two constraints. Your lagrangian is perfectly valid but one finds that it doesnt describe spin 12 particles. Its original prescription rested on two principles. Bibliography 1the quantum theory of fields, volume i foundations, steven weinberg, cambridge university press. We derive the dirac lagrangian density corresponding to the twocomponent dirac. The lagrangian density that gives the dirac equation of motion, i. The rst is naturally associated with con guration space, extended by time, while the latter is the natural description for working in phase space. It is possible to modify this form to get a lagrangian density giving both the wave equation of the particle and the wave equation of the antiparticle.
Lagrangian density article about lagrangian density by the. Hamiltonian density for the dirac equation link to. Formulation of diracs equation start with massless dirac equation. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. It introduces another form for the lagrangian density giving the dirac equation. This is ignored in standard quantum mechanics but is important for what follows. Newton approach requires that you find accelerations in all. Thus, for a scalar field theory the hamiltonian density h is. However, for spinor fields, the magic of the matrices means that the dirac lagrangian is lorentz invariant.
In lagrangian field theory, the lagrangian as a function of generalized coordinates is replaced by a lagrangian density, a function of the fields in the system and their derivatives, and possibly the space and time coordinates themselves. The eulerlagrange equation derived using the fields is the dirac adjoint equation, the hamiltonian density may be derived from the lagrangian in the standard way and the total hamiltonian computed by integrating over space. The lagrangian, lagrangian function or lagrangean l l q, q. Lagrangian formulation of the electromagnetic field thomas yu abstract. An introduction to lagrangian and hamiltonian mechanics. This article uses for the lagrangian density, and l for the lagrangian. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom. Pdf lagrangian density equations of singlefluid and two. Lagrangian the new dirac equation, like the old one see 4 can be considered as an equation of some field. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. The rstorder relativistic oneelectron hamiltonian 49 d.
Results in the differential equations that describe the equations of motion of the system key point. Consequently, lagrangian mechanics becomes the centerpiece of the course and provides a continous thread throughout the text. This is nontrivial because of the noncommutative nature of quaternions. Multiply the non conjugated dirac equation by the conjugated wave function from the left and multiply the. Hamiltons principle states that the motion of a system is such that the action functional s qt zt2 t1 dtlq,q,t. Klauber, student friendly quantum field theory, sandtrove press, 20 chapter 4, problem 4.
This paper will, given some physical assumptions and experimentally veri ed facts, derive the equations of motion of a charged particle in an electromagnetic eld and maxwells equations for the electromagnetic eld through the use of the calculus of variations. Diracs approach was to try to factorize kleingordon equation. Chapter 2 lagranges and hamiltons equations in this chapter, we consider two reformulations of newtonian mechanics, the lagrangian and the hamiltonian formalism. Multiply the nonconjugated dirac equation by the conjugated wave function from the left and multiply the. The lagrangian density for the dirac equation is similar in form 4. In the limit m 0, the dirac equation reduces to the weyl equation, which describes relativistic massless spin 1. It is well known that a free particle with holonomic constraints, imposed by the method of. Claude daviau, in advances in imaging and electron physics, 20. Quantum field theory i babis anastasiou institute for theoretical physics, eth zurich, 8093 zurich, switzerland email. In field theory, the lagrangian density is defined similarly. Lagrangian formalism for the dirac equation including. Derivation of dirac equation using the lagrangian density for. Diracs equation is a model for a electron and positron massive case. Dirac equation, maxwells equation, etc, and cook up a lagrangian that will yield the desired equation of motion.
Classically we have the hamiltonian for a free particle. First that we should try to express the state of the mechanical system using the minimum representation possible and which re ects the fact that the physics of the problem is coordinateinvariant. We derive the diraclagrangian density corresponding to the twocomponent dirac equation. We begin to explore the tensorial densities without derivative, and the momentum. With a corresponding lagrange density function we can obtain the field equation 1. Multiply the nonconjugated dirac equation by the conjugated wave function from the left and multiply the conjugated equation by the wave function from right and subtract the equations. The rst step is to write the dirac equation out longhand. This means to take the righthanded derivative of one of the components, you have to commute it to the right first. Lagrangian density an overview sciencedirect topics. For example, a free, classical electromagnetic field has. The shortest path is an extremal of syx which must, according to the theorems above, obey the eulerlagrange condition. Dirac equation for dummies or theory of elasticity for the seriously advanced james burnett, olga chervova and dmitri vassiliev 30 january 2009 kcl colloquium.
This has to vanish for any x, t, so we obtain the eulerlagrange equation of motion for the. We interpret this as an equation of continuity for probability with j. This lagrangian is complex projected as anticipated in previous articles and this feature is necessary even for a classical real lagrangian. Both the dirac equation and the adjoint dirac equation can be obtained from varying the action with a specific lagrangian density that is given by. The rst is naturally associated with con guration space, extended by time, while the latter is. How can i derive the dirac equation from the lagrangian density for the dirac field. The basic equation of quantum mechanics is the schrodinger equation. The book begins by applying lagranges equations to a number of mechanical systems.
Lagrangian formalism for the dirac equation including charge. Gordon equation describes excitations of a scalar field. Hamiltonian density for the dirac equation 3 the only differences between the two integrals are the minus sign in i d4 and the replacement of jpjin i d4 by min i m4. May 29, 2011 your lagrangian is perfectly valid but one finds that it doesnt describe spin 12 particles. Mechanics is that lagrangian mechanics is introduced in its. In ad dition, the flow equations of a fluid dispersed in a continuum are derived from the ave rage lagrangian density, leading to the classical two fluid model of twophase flow. The problems with the kleingordon equation led dirac to search for an alternative relativistic. Lagrangian densities and principle of least action in. It should be added, however, that it was dirac who found most of the additional insights.
Note that the hamiltonian density is the same as the hamiltonian derived from the dirac equation directly. Lagrangian densities, gravitational field equations and. To leave a comment or report an error, please use the auxiliary blog. This gives us the dirac equation indicating that this lagrangian is the right one. Lagrangian mechanics is a reformulation of classical mechanics, introduced by the italianfrench mathematician and astronomer josephlouis lagrange in 1788 in lagrangian mechanics, the trajectory of a system of particles is derived by solving the lagrange equations in one of two forms. Just as there is no derivation of quantum mechanics from classical mechanics, there is no derivation of relativistic field. The euler lagrange equations, when applied to this lagrangian density, give the dirac equation. The eulerlagrange equation gets us back maxwells equation with this choice of the lagrangian. This lagrangian density is used for all the quarks and leptons only the masses will be different.
Derivation of dirac equation using the lagrangian density. The dirac equation a great deal more was hidden in the dirac equation than the author had expected when he wrote it down in 1928. In klaubers box 43, he uses a symbol with a slash through it to indicate a sum of that symbol multiplied by. It is important to emphasize that we have a lagrangian based, formal classical field theory for electricity and magnetism which has the four components of the 4vector potential as the independent fields. The adoption of quaternions as the base number system in quantum mechanics allows one to define a quaternionic dirac equation in which the wave function is. Dirac equation for dummies or theory of elasticity for the. In 1928 dirac proposed the following form for the electron wave equation.
Lagrangian and dirac constraints for the ideal incompressible. Nesbet 8 has called the lagrangian density 4 seemingly arti. A students guide to lagrangians and hamiltonians a concise but rigorous treatment of variational techniques, focusing primarily on lagrangian and hamiltonian systems, this book is ideal for physics, engineering and mathematics students. The dirac equation can be thought of in terms of a square root of the kleingordon equation. Im studying particle physics from the book of griffiths section 11.
We notice that in the derivation of eulerlagrange equations eq. Lagrangian density article about lagrangian density by. It should be added, however, that it was dirac who found most of the additional. But i think this is explained in nearly every derivation of the dirac lagrangian in books or scripts regarding relativistic quantum mechanics.