The Effective Potential For The Scalar λΦ~4Model At Finite Temperature

Field theory at finite temperature is the fundamental theory in dealing with the nature ofmatter at high temperature and high density. It contains very rich physical phenomenology,spanning fields of nuclear physics, particle physics, astrophysics, condensed physics andcosmology. For example, relativistic degenerate gas formed in a white dwarf star whenordinary matter is greatly compressed, superdense nuclear matter formed in a neutron starwhen the matter is compressed even further so that atomic nuclei overlap, quark gluon plasmaformed in high-energy nuclear collisions, the evolving of spontaneous symmetry breaking ofthe weak and electromagnetic interactions during the big bang. Questions like these haveinterested us very much for a long time. The study of matter under extreme conditions hasdeveloped into a field of interdisciplinary activity and global extent.The study in experiments is mainly high energy heavy ions experiments. One of theimportant goals of the study is to give the phase diagram of the standard model and itsextensions. This phase diagram itself includes very rich phenomenon. So far, one agrees thatthere are at least three distinct phases in the diagram: hadronic phase, quark gluon plasmaphase and color superconducting phase. In the meantime, different parameters will complicatethe phase diagram further. In fact, besides the two parameters of temperature and chemicalpotential, other parameters are possible, such as the number of quarks, the masses of quarks.all of them may affects the phase diagram, even change the order of phase transition. Theintensity of a magnetic field is also an important parameter. In collisions, especially in LHCexperiments, the intensity of the magnetic field can not be ignored. As we all known,magnetic field can affect phase transition.The study of complicated quantum mechanical systems at finite temperature has had asystematic development only in the past few decades. There are now well developed andappropriate formalisms to describe finite temperature field theories. In fact, there are threedistinct, but equivalent formalisms to describe such theories and each has its advantages anddisadvantages. But, the point is that we now have a systematic method of calculating thermalaverages perturbatively in any quantum field theory. We can now study phase transitionsinvolving symmetry restoration in theories with spontaneously broken symmetry. We canstudy deconfinement. At present chiral symmetry restoration phase transition and thedeconfinement phase transition in QCD can be investigated in heavy ion collisionexperiments. This will help us understand properties of the quark-gluon plasma better. Investigating hadron phase transition by using Quantum Chromodynamics at finitetemperature is very difficult, the extent of difficulty can be more serious than the one at zerotemperature. Therefore using effective theories become one of the important research method.The common used models include the chiral perturbative theory, linear sigma models, quarkmeson models and NJL models, they are applied extensively to QCD phase transitionresearch.In this thesis, we will give the effective potential for the scalar4model at finitetemperature. This model is an important model in field theory research. It has a simple formand allows symmetry spontaneous breaking. The related various technology in dealing with itwill form the important technical base for tackling more realistic field theories(such as quarkmeson models and even QCD). We will deduce the effective potential for the4model. Inour work, we will use optimized perturbation theory and expand the effective potential to thesecond order. The optimized perturbation method is essentially a nonperturbation method byusing variational calculation. Since renormalization can even affect the nature of a phasetransition qualitatively, we give the renormalization process in detail. In our calculation, twoand three loops will appear, which correspond to the second and third order of in usualeffective potential calculation respectively. By using the effective potential one caninvestigate further the critical temperature, the order of a phase transition and other quantities.