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Particle Physics is a fundamental theory. Physicists strive to put it on the firmest footing by exploring the best starting points. We will spend some time examining this question.
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Important terms: |
Quantum Field Theory, QFT The typical initial set of problems that are used to illustrate quantum theories describe a single particle. Interactions are treated using potentials. The excited hydrogen atom, for example, is stable in a one particle theory. One needs to allow the possibility of an excited atom as well as a ground state atom and an E&M field or photon. A full blown theory needs to incorporate multiparticle behavior and the evolution of systems where particle number is not a constant (matter-antimatter annihilation and creation). The most promising approach is treating all entities as fields. The state of the art in our understanding of quantum mechanics is multiparticle quantum mechanics as a QFT. |
RENORMALIZATION Discussed some of the difficulties that arise due to the point particle assumption and described how classically one removes the infinite energies associated with a point charge by neglecting the energy required to build the charges. Renormalization is process where several similar infinities can be removed form a QFT |
Standard Model quarks and leptons è elementary building blocks photons, W^{+}, W^{-}, Z_{o}, gluons è interaction quanta much of the underlying structure is generated with local symmetry assumptions internal symmetries |
Symmetry When something doesn’t change when transformed. |
Space-Time symmetries listed above these describe the isotropy of space-time. The laws of physics must conform to certain restrictions in order to preserve these symmetries. |
Internal symmetry Not related to space-time but provides a structure. A good example is isospin. If we consider the proton and neutron to be fundamental and to assume that they are two states of the same particle the nucleon. There are two ways for a nucleon to exist (proton, neutron) but these two states are not related to location or time but reveal some other discernable parameters. |
Continuous and discrete symmetries continuous can be parameterized using a smoothly changing parameter (e.g. _{}) |
Einstein convention _{} |
Let us continue with transformations and classifying objects according to the way they transform.
Review some of the mathematical tools to deal with space and time and also to describe other quantities such as velocity, momentum, force … These quantities will share common structure.
Here are some well know structure
Scalars: temperature, charge, mass, speed
Vectors: position, velocity, Electric field
Tensors: moment of inertia, stress, strain
Elements will have different structure. As a matter of fact the structure can change depending on the theory. For a fully relativistic theory the electric and magnetic fields need to be combined into a second rank tensor and the neither field retains is character as a vector.
Practical tools
1. Matrix representation of rotation
2. Vector notation and vectors
3. Tensors
4. Expanding a charge distribution (tensors)
5. Vector contraction
6. Einstein notation
7. Relativity and four vectors
Point 4: It is very difficult, in general, to find the electric and magnetic fields a point _{} due to a charge distribution _{}
Why is this problem so difficult ?
So physicists ask if there are situations that make the problem more tractable.
If _{} is large compared to _{} [very localized charge distribution at a significant distance from the location where the fields are needed ] one might expect that the details as to how the charge is distributed are unimportant. The dominant feature will be the total charge.
This point of view is mathematically justified by expanding the complete solution in terms of _{} the distance between a point in the charge distribution and the location where the fields need to be evaluated. The problem then separates into a set of terms where each term is characterized by a specific d dependence and a characterization of the structure of the charge distribution as a multipole.
The details of this procedure can be found in most E&M text books and on the web. For example the following sites describe this expansion nicely.
http://en.wikipedia.org/wiki/Spherical_multipole_moments |
http://en.wikipedia.org/wiki/Multipole_expansion |
Usually the student is introduced to this procedure based on the problem expressed in Cartesian coordinates. The more revealing expansion is done in terms of spherical coordinates because the moments can be classified by their transformation properties under rotations.
How would one describe the monopole moment in terms of rotational properties?
{independent è a rotation will not change the monopole moment}
How about the dipole?
{vector è a rotation will mix the three components}
Originally a vector was introduced as a quantity with magnitude and direction. This type of entity was important because so many physics concepts force, position … possess this structure. However another natural way to introduce a vector is to characterize by its transformation characteristics under rotations. There are objects that transform as scalers, vectors, tensors under rotation.
At the beginning of a particle physics course I like to spend some time asking what are the most natural or appealing starting points for a fundamental theory. Historically, one can find examples where the answer was discovered and later a more complete or fundamental justification was developed and other examples where a very thoughtful probe of how things should behave led to a theoretical improvement. There is no correct answer as to the best approach to a fundamental theory but group theory has emerged as a favorite underlying idea.
What properties must physical theories have in order to respect the symmetries of our world?
Waves
Quantum mechanics will develop a new framework for the description of matter and fields. Often the new framework is developed using the familiar concepts of waves and particles of classical physics. Of course, QM being the more fundamental theory, the best way of understanding would be have an intuition at the quantum level as to how matter and energy behave and then extrapolate that to the large scale level. This of course is not really possible because very few people have quantum experience as undergraduates. So we review how classical waves behave. We try to understand in what context they appear classically. The we can use classical wave behavior as a guide but at the same time realize that classical waves and quantum wave-like behavior are not at all the same thing. Classical waves are due to a medium of things that do not behave at all like a wave. Quantum wave-like behavior is just how things behave and all matter will act according to these rules.
Space time
One important ingredient in our theory will be the way we view space and time. For classical theories one basically postulates that our universe exists in something called space that has three dimensions and that events are marked by an absolute quantity called time. Einstein (general relativity) blurred the separation by making space and time a consequence of the evolution of our universe. He also eliminated the separate role that time played as a parameter and linked it with space as a special fourth dimension (special relativity)
Time is absolute (classical)
Distance is absolute (classical)