Energymomentum tensor for a scalar field on manifolds. Stress energymomentum tensor april 19, 2016 1 scalar eld action and associated symmetries a theory of the scalar eld, x can be described by the action, i x d4xl. The energymomentum tensor is the four momentum component in the. Scalar elds, spinor elds, gauge elds, and gravitational elds are treated.
For the canonical energymomentum tensor of nonabelian gauge fields and of. Deriving the energy momentum tensor for a scalar field the energy momentum tensor is defined as the variation of the action with respect to the metric g for inflation we are interested in the action of a scalar field that is given by s z d 4 x vg l z d 4 x vg1 2 g. As with the gradient of a scalar field, if one writes dx as dxe, where e is a unit vector, then in direction grad e a ae dx d 1. The lagrangian remains the same since covariant derivative of a scalar field is the same as normal derivative, i. Similarly, supposing qto be an ndimensional realcomplex vector space. Lecture 10 spin angular momentum, complex poyntings theorem.
Quantum field theory useful formulae and feynman rules. The fourpoint correlation function of the energymomentum. We argue that already at the classical level the energy momentum tensor for a scalar field on manifolds with boundaries in addition to the bulk part contains a contribution located on the boundary. In a complex scalar field theory, the scalar field takes values in the complex numbers, rather than the real numbers. Request pdf renormalization of the energymomentum tensor in noncommutative complex scalar field theory we study the renormalization of dimension four composite operators and the energy. The material is based upon lecture notes for a course i teach from time to time at utah state university on classical field theory. Energymomentum tensor an overview sciencedirect topics. We present an explicit momentum space computation of the fourpoint function of the energy momentum tensor in 4 spacetime dimensions for the free and conformally invariant theory of a scalar field. Energymomentum tensor for a scalar field on manifolds with. Pdf boson stars in general scalartensor gravitation.
The complex scalar lagrangian has a symmetry under. The energy momentum tensor is the most universal operator forconformal. Vacuum expectation value of the surface energymomentum. On the energymomentum tensor of the scalar field in. Complex scalar eld we want to investigate the theory of a complex scalar eld. But since the conjugate transpose of a real scalar number is itself, we have, if h h is real, then h hy h h where yimplies conjugate transpose. The above, in detail, using the rule of matrix algebra.
For electromagnetism we have not a scalar but four fields. Lagrangian with respect to the gauge field and for the energymomentum. Currents and the energymomentum tensor in classical field theory. The renormalized stress tensor of a free quantum field in a curved spacetime should be obtainable by some covariant procedure from the shortdistance behavior of the corresponding operator products evaluated at separated points. This extensions showed the possibility of classify the cosmological behaviour of scalar tensor theories in equivalence sets, where the field itself is a class variable. We study the dynamical description of gravity, the appropriate definition of the scalar field energy momentum tensor, and the interrelation between them in scalar tensor theories of gravity. On the energymomentum tensor in moyal space springerlink. On the question of the uniqueness of the energymomentum. Suppose we have a continuous transformation, which is internal, that transforms the fields according to.
In the theory of relativity, this momentum vector is taken as the four momentum. We can generalize the formalism to complex scalar fields. We also consider a direct coupling the scalar field to the ricci scalar curvature which is parameterized by an. Similarly, supposing q to be an ndimensional real complex vector space. The stress energy tensor is defined as the tensor t of order two that gives the flux of the. We work in the mostlyminus convention for the minkowski metric 1. Another wellknown example of a scalar field theory with kink solutions is the sinegordon theory. Classical field theory is introduced in chapter 3 and a careful treatment of the noether theorem and the energy momentum tensor are given. On the energymomentum tensor of the scalar field in scalar. The stress energy tensor, sometimes called the stress energy momentum tensor or the energy momentum tensor, is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of newtonian physics. Compared with our previous researches, this action provides the possibility to explore the energy momentum tensor and sound speed of the single field action of gravity. In an astrophysical setting, if a scalar tensor theory describes gravitation, the value of the effective gravitational constant far out from the star must not necessary be the. I argue that according to general relativity t is not an intrinsic property of matter, looking at how the energy tensor for a relativistic material system can be derived in a lagrangian framework.
Lecture 17 april 1, 2010 the stress energymomentum tensor. Phantom and quintessence fields coupled to scalar curvature. The matter field depends on one scalar quantity and one vector quantity. One of the results of the lagrangian variation is the stress energy tensor of the pressure field 2. Aug 16, 2004 we show that the energy momentum tensor in the noncommutative complex scalar field theory is defined up to six arbitrary constants. In general relativity, the stress energy tensor is symmetric. Proof that energy momentum tensor of scalar field theory.
Energy momentum tensor of a scalar field by varying the. The vacuum expectation value vev of the surface energymomentum tensor is proportional to the vev of field squared cosmological constant type contribution on the brane in 1brane model regularized quantities are obtained via zeta function regularization, with the help of extended chowlaselberg formula. It is an attribute of matter, radiation, and nongravitational force fields. With these formulae, one nds in the case of a scalar eld with action 10. The yangmills stressenergy tensor and force equation. Based on noethers theorem, construct the energy momentum tensor for classical electro. This is the energy momentum tensor for the field a. Pdf currents and the energymomentum tensor in classical. As seen previsouly, t tt represents the density of relativistic mass, i. We are given a lagrangian density lx and a global symmetry transformation, parametrised by which will be a. The canonically defined energy momentum tensor is not finite and must be replaced by the improved one, in order to provide finiteness. All contributions are manifestly positive definite assuming v is bounded below.
One of the simplest energy momentum tensors is the dust energy momentum tensor. Note that the usual procedure does not result in a symmetric tensor. The principle of local gauge invariance since this is a course in eld theory, we are required to only use elds to model things like electrically charged matter, so we insist upon a model for the charged sources built from a classical eld. Pdf currents and the energymomentum tensor in classical field. We work by embedding the scalar field theory in a gravitational background consistently with conformal. In particular, it is found that the energy density, as defined in the previous paper, is identical with that given by the physical energy momentum tensor. Momentum operator of the quantized real kleingordon field. Schroeder, addisonwesley 3quantum field theory in a nutshell, a. Complex scalar klein gordon field coupled to electro.
The above discussions only applied to operators, let us nally discuss. Jan 27, 2017 this is the energy momentum tensor, also known as the stress energy tensor for the dust. In particular, a dirac eld describes charged fermions while a majorana eld describes neutral fermions. Despite of this argument, one usually notices that it is desirable, nevertheless, to construct at least an \e ective gravitational energy momentum tensor. Spin angular momentum, complex poyntings theorem, lossless condition, energy density97 that they are also scalar numbers. The ksf coupling is algebraically different between matter, em field, and scalar field a departure from standard scalar tensor theory. Energy momentum tensor of a massless scalar quantum field in. The result is obtained by explicit evaluation of the feynman diagrams by tensor reduction. Therefore, equation 5 is a new action with gravity, phantom and quintessence fields and scalar curvature r. Similarly, supposing q to be an ndimensional realcomplex vector. This conserved current is called the energymomentum tensor. However, scalar quantum fields feature in the effective field theory descriptions of many physical phenomena. The stressenergy tensor, sometimes called the stress energy momentum tensor or the energy momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of newtonian physics. We study the renormalization of dimensionfour composite operators and the energy momentum tensor in noncommutative complex scalar field theory.
Qauntum field theory quantizing the complex scalar january 9, 2018 1 quantizing the complex scalar field we will analyze the qft of a free complex scalar. Energymomentum tensor and effective lagrangian of scalar qed. Contents 1 from classical theory to quantum theory 2 quantization of real scalar eld 3 quantization of complex scalar eld 4 propagator of kleingordon eld 5 homework wei wangsjtu lectures on qft 2017. Lorentzpoincare covariant formulation of field theory. Pdf energymomentum tensor and effective lagrangian of. Quantum field theory example sheet 1 michelmas term 2011. Canonical quantization of truncate the complex scalar field. Using the standard variational procedure for the action with the boundary term, the expression for the surface energy momentum tensor is derived for arbitrary bulk and boundary geometries. In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. Sargsyan department of physics, yerevan state university. Deriving the energy momentum tensor for a scalar eld. It has been known for a long time that conformal invariance completely. This energymomentum tensor is identical to the one obtained from noethers theorem.
A scalar field is invariant under any lorentz transformation the only fundamental scalar quantum field that has been observed in nature is the higgs field. We show that the quantity which one would naively identify as the energy momentum tensor of the scalar field is not appropriate. The transition to classical field theory proceeds via the replacements qit. We can check explicitly that the lagrangian density is invariant under this transfo. This object is called energy momentum or stress energy tensor. Energymomentum tensor and effective lagrangian of scalar. Both the ksf and the electromagnetic em field energy momentum in contribute to curvature of the 4d metric. Classical field theory and the stress energy tensor iopscience. Consider the lagrangian density of a free real scalar field. The procedure of finding the stressenergy tensor and. Qauntum field theory quantizing the complex scalar. Therefore,itscorrelation functions are natural objects to study in any cft. This type of matter field consists of noninteracting incoherent matter.
Free complex scalar field 7 quantizing the elds gives. Quantum field theory department of applied mathematics and. The general canonical energy momentum tensor is defined by these components. This expression is found by substituting 0 for the indices, and the explicit minkowski form o the metric tensor. Field equations and lagrangian of the kaluza energy. On the energymomentum tensor for a scalar field on manifolds with. After covering real and complex scalar fields, the author impressively introduces the dirac spinor via the weyl spinor. Grishchuk 12 1 department of physics and astronomy, cardi university, cardi cf24 3yb, uk. Quantum field theory i eth zurich, hs12 chapter 4 prof. The energymomentum tensors in classical gauge theories. Problems in quantum field theory of fundamental interactions. The stress energy tensor, sometimes called the stress energy momentum tensor or the energy momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of newtonian physics.
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