First Principles of Particle Physics and the Standard Model
Now that we’ve gone through quite a bit of mathematical formalism we’re finally ready to start talking about physics. So that it’s clear what we’re doing, we’ll state our goal up front: we want to formulate a relativistic quantum mechanical theory of inte
- PDF / 1,218,291 Bytes
- 127 Pages / 439.37 x 666.142 pts Page_size
- 15 Downloads / 260 Views
First Principles of Particle Physics and the Standard Model
4.1 Quantum Fields Now that we’ve gone through quite a bit of mathematical formalism we’re finally ready to start talking about physics. So that it’s clear what we’re doing, we’ll state our goal up front: we want to formulate a relativistic quantum mechanical theory of interactions. If any of those three components (relativity, quantum, interaction) is missing we’ll need to keep working. Our approach will be to start with what we know, see what is lacking, and then try to incorporate whatever is needed. So, we’ll start with the fundamental equation of quantum mechanics, Schroedinger’s equation,1 H ‰ D i„
@‰ : @t
We know that for a non-interacting, non-relativistic particle, H D
(4.1) p2 2m
„ D 2m r 2 , so
„2 2 @‰ r ‰ D i„ : 2m @t
2
(4.2)
For our purposes, we want to emphasize something that is usually not emphasized in introductory quantum courses – that ‰ is a Scalar Field. The physical meaning of this is that it only describes a particle with a single state. The way we interpret this is that it describes a particle with spin 0. Or in the language we learned about in the previous sections, it sits in a j D 0 representation of SU.2/, which is the trivial representation. A particle in a larger representation (i.e. a j D 12
1
If you’re rusty on quantum mechanics at this point don’t worry. We will talk more about what quantization is in Sect. 4.6. Now we’re just providing motivation for the field equations we’ll be using in this section.
M. Robinson, Symmetry and the Standard Model: Mathematics and Particle Physics, DOI 10.1007/978-1-4419-8267-4 4, © Springer Science+Business Media, LLC 2011
141
142
4 First Principles and the Standard Model
representation) can be in multiple states (i.e. C 12 or 12 ) and therefore requires multiple components. A scalar obviously can’t have multiple states. Furthermore, ‰ doesn’t have any spacetime indices, so it also transforms trivially under the Lorentz group SO.1; 3/. However, despite this lack of difficulty with Lorentz transformations, we do have a fundamental problem in making this a truly relativistic theory. Namely, the spatial derivative in (4.2) acts quadratically (r 2 ), whereas the time derivative is linear. Treating space and time differently in this way is unacceptable for a relativistic theory, because relativity requires that they be treated the same. The reason for this is that if space and time are to be seen simply as different components of a single geometry, we can no more treat them differently than the x and y dimensions can be treated differently in Euclidian space. This is an indication of a much deeper problem with quantum mechanics: quantization involves “promoting” space from a parameter to an operator, whereas time is treated the same as in classical physics – as a parameter. This fundamental asymmetry is what ultimately prevents a straightforward generalization to a relativistic quantum theory. So to fix this problem, we have two choices: either promote time to an operator
Data Loading...
