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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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Calculus Of Variations

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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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