{"id":20,"date":"2018-11-13T06:15:20","date_gmt":"2018-11-13T06:15:20","guid":{"rendered":"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=20"},"modified":"2019-04-30T09:11:27","modified_gmt":"2019-04-30T09:11:27","slug":"classical-field-theory","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/chapter\/classical-field-theory\/","title":{"rendered":"Classical Field Theory"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
\r\n\r\n\u00a0<\/strong>\r\n\r\n \r\n\r\n \r\n\r\n \u00a0 1.<\/strong>\u00a0Learning Outcomes<\/strong>\r\n\r\n<\/div>\r\n
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    \r\n \t
  1. Become aware of Particle mechanics in Lagrangian and Hamiltonian forms. Alternative to starting with equations of motion.<\/strong><\/li>\r\n \t
  2. Get to know the Variational principle for deriving the equations of motion.<\/strong><\/li>\r\n \t
  3. Get to know the eqations of motion in Lagrangian and Hamiltonian forms.<\/strong><\/li>\r\n \t
  4. Learn how one moves away from finite degrees of freedom to a field, which is a system with infinite degrees of freedom.<\/strong><\/li>\r\n \t
  5. Become familiar with the notation for covariant and contravariant vectors and tensors.<\/strong><\/li>\r\n \t
  6. Learn the summation convention.<\/strong><\/li>\r\n \t
  7. Derive Klein Gordon equation from the scalar Lagrangian.<\/strong><\/li>\r\n<\/ol>\r\n\u00a0 \u00a0 2.\u00a0<\/strong>Introduction<\/strong>\r\n

    Quantum mechanics applies quantum ideas to classical mechanics of a classical particle, like an atom. We also have classical fields like the electromagnetic fields (e.m.field). How do we quantize the fields. This study goes under the\u00a0name of Quantum Field Theory (QFT). The most\u00a0<\/span>familiar field, electromagnetic field<\/span>,when studied in a quantized form is called quantum electrodynamics (QED).<\/span><\/p>\r\n \r\n

    In the <\/span>mid 1950s a reasonably consistent theory of QED was obtained. This enabled the consistent handling of divergences (or <\/span>infinites) that occurred in the theory. The successful extension of those ideas to non-abelian gauge theories and to the unification of weak and <\/span>electro magnetic interaction took another 25 years. This resulted in the award of the Nobel prize to Glashow, Weinberg <\/span>and Salam in the year 1979.<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n

    In this course we are interested in studying the emission, absorption and scattering of light in Quantum Electro Dynamics. The early beginnings in 1928-30, were called Dirac\u2019s theory of radiation; hence the title of this course. To quantize we first identify the degrees of freedom of the field and their conjugates using the Largrangian formulation. We then impose the commutation relations between the field degrees of freedom and its conjugate. In this module we formulate the classical field theory .<\/p>\r\n \r\n\r\n3.\u00a0<\/strong>Particle mechanics in Lagrangian form.<\/strong><\/span>\r\n\r\n \r\n\r\n3.1 Variational Principle<\/strong><\/span>\r\n\r\n \r\n

    Instead of starting with equation of motion namely the second law of Newton we can use the principle of least action to derive the equation of motion. In this way we are making our starting point of the subject more broad based. This has several advantages\u00a0that we will see as we proceed. This formulation makes the transition to quantum mechanics much easier.<\/span><\/p>\r\n \r\n

    Non-relativistic particles obeying Newton\u2019s law satisfy <\/span>a second order\u00a0<\/span>differential equation. If we know the initial values of the position and the velocity of the\u00a0<\/span>particle and the force or potential energy we can calculate its complete trajectory. Let\u00a0??<\/sub>(t<\/span>) and ?.<\/sup>?<\/sub>(t) be the set of\u00a0 coordinates and velocities respectively which describe the\u00a0<\/span>configuration and evolution of the system.\u00a0 We use the notation where a dot on top\u00a0<\/span>denotes time derivative. We define a mathematical object S as Action of the system by\u00a0<\/span>the equation.<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n

    Where t1<\/sub> and t2<\/sub> are initial and final times between which we are studying the evolution of the system. If there is a source or sink of energy the Langragian can also depend explicitly on time. Otherwise it depends only on coordinates qr<\/sub> and velocities q\u0307r<\/sub>. \u2018Principle of Least Action or Varational Principle\u2019 says that among all the trajectories<\/p>\r\n\"\"\r\n

    where the summation convention for index r is used. We assume there is no explicit time dependence. Integrating by parts we get\u00a0since we are looking at given initial and final points\u00a0??<\/sub> (?1<\/sub>) and ??<\/sub> (?2<\/sub>) we take\u00a0<\/span>??<\/span>?<\/sub> (?<\/span>1<\/sub>) = ??<\/span>?<\/sub> (?<\/span>2<\/sub>) = 0 Stationary implies that ?? = 0 vanishes. Since this is true for any arbitrary variation of the path i.e. for <\/span>arbitrary , ??? , ?? = 0 implies<\/span><\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\n\u00a0 \u00a0 These are the equations of motion for the coordinates also known as Euler-Lagrange equations.\r\n\r\n \r\n

    For a particle of mass moving in a time independent potential ?(?) Lagrangian L usually defined as ? = ? \u2212 ? = Kinetic Energy \u2013 Potential Energy.<\/p>\r\n\"\"\r\n\r\nas expected from Newton\u2019s second law.\r\n\r\n \r\n\r\n3.2 Hamilton formalism and Poisson Brackets.<\/strong>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\n\u00a0 \u00a0\u00a0\"\"\r\n\r\n \r\n\r\n4.\u00a0<\/strong>Action principle for a field theory.<\/strong><\/span>\r\n\r\n \r\n\r\n4.1 Field as a system with infinite degrees of freedom<\/strong><\/span>\r\n\r\n \r\n

    When the degrees of freedom are continuous ,their number tends to infinity and the system is called field . For a field\u00a0? we can write the Lagrangian as an integral<\/p>\r\n \r\n

    ? = \u222b ?3<\/sup>? \u2112\u2329?(?), ??<\/sub>?(?)\u232a\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(15)<\/p>\r\n \r\n\r\n\u2112\u00a0 is the lagrangian density and it depends on the space derivatives, besides the time derivative or the velocities. The action S takes the form\r\n\r\n \r\n

    ? = \u222b ??? = \u222b ?? ?3<\/sup>? \u2112\u2329?(?), ??<\/sub>?\u232a\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (4.1)<\/p>\r\n \r\n

    We will refer to the Largrangian density as just Lagrangian and L as total Lagrangian. The field is a set of numbers at each point in space. The Lagrangian is a\u00a0function of space and time as it depends on field ? at each point and its derivatives.\u00a0 The action S is a number. It is a rule that associates a real number with a given function . If field configuration\u00a0? changes the number also changes. Such objects are called funtionals. The action is a functional of the field. (or fields if this is more than one field.) Total lagrangian L is a function of t but a functional of the fields at a given t.<\/p>\r\n \r\n\r\n4.2 Covariant Notation.<\/strong>\r\n\r\n \r\n\r\nWe take units in which c= 1 and \u0127 \u225d h\/2\u03c0 =1.\r\n\r\n<\/div>\r\n

    <\/div>\r\n
    ?? = (?0<\/sup>, ?1<\/sup>, ?2<\/sup>, ?3<\/sup>) \u2261 (?, ?, ?, ?)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(4.2)<\/div>\r\n
    <\/div>\r\n
    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 where ?0<\/sup> = ?? = ? (if ? = 1)<\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n \r\n

    We assume the Summation Convention:\u00a0 Repeated indices are summed. Greek indices go over 0, 1, 2, 3. Latin indices go over the space indices 1, 2, 3 only. One of the repeated indices must be in the lower and the other in upper position, (otherwise there has been a mistake)<\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\n \r\n\r\n4.3 Euler Lagrange equation.<\/span><\/strong>\r\n\r\n \r\n

    The principle of least action states that when a system evolves from one configuration to another between times t1 and t2 it takes a path or evolves through field configuration for which action S is stationary (usually a minimum).<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\n\u00a0 \u00a0\u00a0\"\"\r\n\r\n \r\n\r\n4.4 Hamiltonian Formalism.<\/strong><\/span>\r\n\r\n \r\n\r\nThe momenta canonical to the field are defined similar to the case of finite degrees of freedom as\r\n\r\n\"\"\r\n\r\n \r\n\r\n4.5 Scalar field and Klein Gordon equation of motion<\/span><\/strong>\r\n\r\n \r\n\r\nThe Lagrangian for the scalar field is\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\nThe Lagrangian was chosen to give the K.G. equation.\r\n\r\n \r\n\r\n5. Summary<\/strong>\r\n\r\n \r\n

    We have learnt how to derive equations of motion for classical field theory starting with a variation or action principle.<\/p>\r\n \r\n

    We have learnt the notation for space time metric, for the covariant and contravariant tensors, for the product of tensors and the summation convention .<\/p>\r\n \r\n

    We use the Lagrangian function which is kinetic energy minus the potential energy in classical mechanics. In field theory it is constructed using general arguments like symmetries. It should lead to the known equation of motion.<\/p>\r\n \r\n

    We also defined the Hamiltonian function which is the total energy. We have seen that the equation of motion can be formulated in terms of either Lagrangian or Hamiltonian, apart from being based on Newtonian dynamics, the second law. They are equivalent\u00a0but we will see that the Lagrange \u2013 Hamiltonian form makes <\/span>transition to quantum mechanics easier.<\/span><\/p>\r\n \r\n

    With a suitably defined Lagrangian for the scalar <\/span>field we obtain the Klein \u2013 Gordon equation as the equation of motion.<\/span><\/p>\r\n \r\n

    5.3<\/strong> Self Learn<\/strong><\/p>\r\n \r\n

    Key Concepts:<\/span><\/p>\r\n \r\n

    Equation of motion,\u00a0 Newton\u2019s law, Acceleration, Force.<\/span><\/p>\r\n \r\n

    Potential energy, Kinetic Energy.<\/span><\/p>\r\n \r\n

    Lagrangian,<\/span><\/p>\r\n \r\n

    Hamiltonian,<\/span><\/p>\r\n \r\n

    Poisson Brackets<\/span><\/p>\r\n \r\n

    Degrees of Freedom. Finite and Infinite.<\/span><\/p>\r\n \r\n

    Differentiation,\u00a0 Integration by parts, Surface and volume integrals.<\/span><\/p>\r\n \r\n

    Gauss\u2019s theorem<\/span><\/p>\r\n \r\n

    Four dimensional integrals.<\/span><\/p>\r\n \r\n

    Starting with <\/span>equation of motion.\u00a0 \u2013 Newton\u2019s Law<\/span><\/p>\r\n \r\n

    Starting with Variation or Action Principle.<\/span><\/p>\r\n \r\n

    Scalar field and <\/span>Klein Gordon equation.<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\u00a0 \u00a0 5.4<\/strong>\u00a0 <\/strong>Self assessment<\/strong>\r\n\r\n<\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\n5.5<\/strong> Learn More<\/strong>\r\n\r\n \r\n\r\nBooks<\/strong>\r\n