- first clues on invariance (how or why did invariance and group formalisms take such a central position in qft discourse?)
- but not on gauge or other problems like decoherence.
- in the current formalism (in the language of group theories (and what else?)) , quantities and formula (equations) are looked at
from the perspective of invariance.
- This seems to have arisen from a categorical treatment of linear transformations to determine or work correctly with the "cannonical components" (p1,…,pn,q1,…,q2) in the 2N phase space of the Hamiltonian formalism.
- This treatment of linear transformations is such as (some kind of ) of operator analysis
- that involves things like commutators (cf. Shankar's commutators discussion in linear vector spaces, ch. 1)
- Poisson brackets, very similar to commutators , as in they have each an exactly similar set of three identities or properties (defining what looks like the same algebra).
- related notes: the cross product , like the dot product depends on the metric of the space , unlike dp it also depends on the handedness of the space or coord system. note also the antisymmetry in axb=-bxa reminiscent of h/skew-symmetry of inner products.
- constants of motion - these must be like the eigenfunctions of the functions (dependent variables, xforms) in the commutators or Poisson brackets?
- discussion of constants of motion in [cohen]
- A result in the Poisson bracket treatment is
- since operators are linear xforms and in turn just functions , variables are considered that are dependent on the canonical coords. , ie,
- f(q,p) , g(p,q)
- note these are note necessarily explicitly dependent on time, though some treatments use f(q,p,t).
- Without explicitly defining it here, the way the poisson bracket is defined
sets it equal to the time derivative (rate of change wrt time) of one of the variables (or measurables) - Thus if the Poisson bracket of a given variable "vanishes" it means the variable remains constant.
- because variables (aka functions, lin. xforms, operators, measurables, observables) that are constant have zero-valued derivatives)
- df/dt = 0 ⇒ f = k , k constant.
- Thus a zero valued Poisson bracket represents an invariance of symmetry of the transformation or variable.
- This is the first reasoned trek from talking about linear transformations and how they affect computed measurables to talking about a symmetry or invariance.
- Indeed in Fields Siegel opens his discussion of symmetry with commutators and brackets.
- He writes:
<pre> "In the Hamiltonian approach to mechanics, both symmetries and dynamics can be expressed conveniently in terms of a \bracket": the Poisson bracket for classical mechanics, the commutator for quantum mechanics. In this formulation, the fundamental variables (operators) are some set of coordinates and their canonically conjugate momenta, as functions of time. The (Heisenberg) operator approach to quantum mechanics then is related to classical mechanics by identifying the semiclassical limit of the commutator as the Poisson bracket: For any functions A and B of p and q, the quantum mechanical commutator" </pre>
– ([Siegel] ch. 1.A.1 "nonrelativity")
- This still does not motivate with any clarity the adoption of group theoretic techniques; except for two things: a trivial observation and something gleaned from the day's review of several QFT books (added within the past day to the references.phys) so far.
- The trivial observation:
- one thing i can think of as to why use group techniques is that formally, axiomatically, analytically, categorically, algebraically, groups are a generalization of vector spaces and spaces of linear transformations on those vector spaces. Indeed all vectors spaces (including those of linear xforms) are groups.
- (the) Something gleaned so far: (stands to be corrected big time nohokuku mazalekya
)- The use of invariance classes (such as commutators , Poisson brackets, Lie brackets, or groups) to study analyse or compute operators (lin. transformations) greatly simplifies the recalculations necessary to account for changes in coordinate frames , which requires a set of spatial transformation applications on the systems being analyzed.
- this abstracts or sublimates the operations required to make correct computations for all possible transformations (mutations in reference frames)
- it does so by considering formalisms with only those quantities that are left unchanged by the transformations. This organizes the transformations into classes for which those observables remain constant. The transformation groups then become the symmetry groups for a given quantum field equation which is also called a field theory, or rather a given field's theory.
- later on in the discourse gleaned from texts skimmed earlier ,
- we see the uses of homomorphisms from group transformation spaces to vector spaces (group representations), or perhaps also,
- spaces of such homomorphisms (representations) defined on things other than fields, namely rings, ie, being not linear spaces, but modules.
- (since linear vector spaces are defined only on fields (of complex or real scalars).)
- it is worth noting also that
- the bracket is like a delta function (eg a kroenecker delta) cf. [Siegel] p. 4.
- ie, similar to or is a metric, a sort of distance
- ie, a norm
- hence the speaking norms and seminorms , and hence gauge - since the gauge is "a seminorm" - cf. gauge theory, Norm_(mathematics),
- Hence when we speak of variables and operators that are invariance in the Poisson Bracket , or for which the Poisson Bracket vanishes, we are speaking of the invariance of bracket for that variable, thus a metric,distance,norm,seminorm or gauge invariance.
- both both previous notes: there's a seed for a discussion on gauge as a seminorm taken from a text on norms at wp. (op. cit.)
