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CHAPTER 2. LAGRANGIAN QUANTUM FIELD THEORY §2.1 GENERAL FORMALISM In quantum field theory we will consider systems with an infinite number of quantum mechanical dynamical variables. As a motivation and guide for the many definitions and procedures to be discussed, we will use concepts and techniques from classical field theory, the classical mechanics of infinitely many degress of freedom. The quantum fields will in addition become operators in Hilber
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  CHAPTER 2.LAGRANGIAN QUANTUM FIELD THEORY § 2.1 GENERAL FORMALISM In quantum field theory we will consider systems with an infinite number of quantum mechanical dynamical variables. As a motivation and guide for the manydefinitions and procedures to be discussed, we will use concepts and techniquesfrom classical field theory, the classical mechanics of infinitely many degress of freedom. The quantum fields will in addition become operators in Hilbert spaceand, as we will see, not every “classical” manipulation will make sense. We willhave to discover the modifications to our theory needed to define a consistentquantum field theory in a sort of give and take process. We begin by recalling thebasic tennants of classical field theory. In general we will consider a continuoussystem described by several classical fields  φ r ( x ) ,r  = 1 , 2 ,...N.  (In general we willdenote classical fields by lower case letters and quantum fields by the upper caseof the same letters.) The index r can label the different components of the samefunction such as the components of the vector potential   A ( x ) or the index canlabel two or more sets of completely independent fields like the components of thevector potential and the components of the gravitational field  g µν  ( x ). Also  φ r ( x )can be complex, in which case,  φ r  and  φ ∗ r  can be considered independent or thecomplex fields can be written in terms of real and imaginary parts which then canbe treated as independent. The dynamical equations for the time evolution of thefields, the so called field equations or equations of motion, will be assumed to bederivable from Hamilton’s variational principle for the action S  (Ω) =   Ω d 4 x L ( φ,∂  µ φ r ) (2 . 1 . 1)where Ω is an arbitrary volume in space-time and  L  is the Langrangian densitywhich is assumed to depend on the fields and their first dervatives  ∂  µ φ r . Hamil-ton’s principle states that S is stationary δS  (Ω) = 0 (2 . 1 . 2)under variations in the fields φ r ( x )  → φ r ( x ) + δφ r ( x ) =  φ  r ( x ) (2 . 1 . 3)80  which vanish on the boundary  ∂  Ω of the volume Ω, δφ r ( x ) = 0 on  ∂  Ω .  (2 . 1 . 4)The physical field configuration in the space-time volume is such that the action Sremainsinvariant undersmallvariationsin the fieldsfor fixed boundaryconditions.The calculation of the variation of the action yields the Euler- Lagrange equationsof motion for the fields δS  (Ω) =   Ω d 4 xδ  L =   Ω d 4 x   ∂  L ∂φ r δφ r  +  ∂  L ∂∂  µ φ r δ∂  µ φ r  .  (2 . 1 . 5)But δ∂  µ φ r  =  ∂  µ φ  r  − ∂  µ φ r =  ∂  µ ( φ  r  − φ r ) =  ∂  µ δφ r .  (2 . 1 . 6)Thus, performing an integration by parts we have δS  (Ω) =   Ω d 4 x   ∂  L ∂φ r − ∂  µ ∂  L ∂∂  µ φ r  δφ r +   Ω   ∂  L ∂∂  µ φ r ( ∂  µ δφ r ) + ∂  µ ∂  L ∂∂  µ φ r δφ r  =   Ω d 4 x   ∂  L ∂φ r − ∂  µ ∂  L ∂∂  µ φ r  δφ r  +   Ω d 4 x∂  µ [  ∂  L ∂∂  µ φ r δφ r ] .  (2 . 1 . 7)The last integral yields a surface integral over  ∂  Ω by Gauss’ divergence theorem   Ω d 4 x∂  µ F  µ =   ∂  Ω d 3 σ  µ F  µ (2 . 1 . 8)but  δφ r  = 0 on  ∂  Ω hence the integral is zero. So δS  (Ω) =   Ω d 4 x   ∂  L ∂φ r − ∂  µ ∂  L ∂∂  µ φ r  δφ r  = 0 (2 . 1 . 9)by Hamilton’s principle. Since  δφ r  is arbitrary inside Ω, the integrand vanishes ∂  L ∂φ r − ∂  µ ∂  L ∂∂  µ φ r = 0  for r  = 1 ,...,N.  (2 . 1 . 10)81  These Euler-Lagrange equations are the equations of motion for the fields  φ r .According to the canonical quantization procedure to be developed, we wouldlike to deal with generalized coordinates and their canonically conjugate momentaso that we may impose the quantum mechanical commutation relations betweenthem. Hence we would like to Legendre transform our Lagrangian system to aHamiltonian formulation. We can see how to introduce the appropriate dynamicalvariables for this transformation by exhibiting the classical mechanical or particleanalogue for our classical field theory. This can be done a few ways, in the intro-duction we introduced discrete conjugate variables by Fourier transforming thespace dependence of the fields into momentum space then breaking momentumspace into cells. We could also expand in terms of a complete set of momentumspace functions to achieve the same result. Here let’s be more direct and work inspace-time. For each point in space the fields are considered independent dynam-ical variables with a given time dependence. We imagine approaching this con-tinuum limit by first dividing up three-space into cells of volume  δx i ,  i  = 1 , 2 , ··· .Then we approximate the values of the field in each cell by its average over thecell¯ φ r ( x i ,t ) = 1 δx i   δ x i d 3 xφ r ( x,t )  ≈ φ r ( x i ,t ) .  (2 . 1 . 11)This is roughly the value of   φ r ( x,t ) say at the center of the cell  x  =  x i  .Our field system is now is described by a discrete set of generalized coordi-nates, q  ri ( t ) = ¯ φ r ( x i ,t )  ≈ φ r ( x i ,t ) ,  (2 . 1 . 12)the field variables evaluated at the lattice sites, and their generalized velocities˙ q  ri ( t ) = ˙¯ φ r ( x i ,t )  ≈  ˙ φ r ( x i ,t ) .  (2 . 1 . 13)Since  L  depends on     φ r  also, we define this as the difference in the field valuesat neighboring sites. Thus,  L ( x,t ) in the  i th cell, denoted by  L i , is a function of  q  ri ,  ˙ q  ri  and  q   ri  the coordinates of the nearest neighbors, L i  = L i ( q  ri ,  ˙ q  ri ,q   ri ) .  (2 . 1 . 14)Hence, the Lagrangian is the spatial integral of the Langrangian densityL( t ) =    d 3 x L =  i δx i L i ( q  ri ,  ˙ q  ri ,q   ri ) .  (2 . 1 . 15)82  and we have a mechanical system with a countable infinity of generalized coordi-nates.We can now introducein the usualway the momenta  p ri  canonically conjugateto the coordinates  q  ri  p ri ( t ) =  ∂L∂   ˙ q  ri ( t ) =  j δx j ∂  L j ( t ) ∂   ˙ q  ri ( t ) =  δx i ∂  L i ( t ) ∂   ˙ q  ri ( t ) (2 . 1 . 16)and the Legendre transformation to the Hamiltonian is H  ( q  ri ,p ri ) =  H   =   p ri  ˙ q  ri − L =  i δx i  ∂  L i ( t ) ∂   ˙ q  ri ( t ) ˙ q  ri ( t ) −L i  .  (2 . 1 . 17)The Euler-Lagrange equations are now replaced by Hamilton’s equations ∂H ∂q  ri = − ˙  p ri  , ∂H ∂p ri = ˙ q  ri  .  (2 . 1 . 18)We define the momentum field canonically conjugate to the field coordinate by π  r ( x i ,t ) =  ∂  L i ( t ) ∂   ˙ q  ri ( t ) =  ∂  L i ∂   ˙ φ r ( x i ,t ) .  (2 . 1 . 19)Then  p ri ( t ) =  π  r ( x i ,t ) δx i H   =  i δx i [ π  r ( x i ,t ) ˙ φ r ( x i ,t ) −L i ] .  (2 . 1 . 20)Going over to the continuum limit  δx i  →  0 the fields go over to the continuumvalues  φ r ( x,t )and the conjugate momentum fields to theirs  π  r ( x,t ) (recall  π  r  is afunction of   φ r ,∂  µ φ r ) and  L i  →L . So that π  r ( x ) =  ∂  L ∂   ˙ φ r ( x ) .  (2 . 1 . 21)The Hamiltonian corresponding to the Lagrangian L is H   =    d 3 x H  (2 . 1 . 22)83
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