interaction/particle groups in a math object context

27.x.XXX

This graph shows a few of the groups describing particle families based on fundamental interaction (force) fields in which they figure as gauge particle, or field generator / interactors (eg, the standard model aiming to include all particles, involves the cross product of three groups - for three fundamental interactions - note there is none for gravity, as no particle for it is yet known :)), in relation to other mathematical objects, including finite automata (computers).


It is reproduced from this earlier post. Made using Graphviz/2.22 :)

###

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how cometh dept: ortho fns that dont cancel out

13.x.1430

in [Waerden]** §3 :

*
Lin transformation product with scalar product of two orthogonal functions.
o
<φ_j,Aψ_k> = A<φ_j,ψ_k>
*
The scalar product should be zero b/c the two functions φ,ψ are orthogonal
*
And yet van der Waerden states this is what gives the coeffs of A:
o
<φ_j,Aψ_k> = A<φ_j,ψ_k> = a_jk
*
¡¡¡ How Cometh ?!!¿؟?



** [waerden] = van der Waerden, Group theory and quantum mechanics , Springer 1974.
English translation of Die gruppentheoretische methode in der quantenmechanik , Berlin: Springer Verlag, 1932.


. ... Read more

social fields

27.ramadan.MCDXXX

updated version of an earlier post with some additions.

*
we imagine a human society where each individual is a generator of at least one field, or is the gauge particle of a field
*
likewise we can imagine the individual comprised a bunch of gauge charges (flavors) each interacting / generating different fields - just like a particle could interact strongly, e-weakly and gravitationally

*
or rather that every individ is both a gauge particle generating a field, and endowed with different charges , "observables that interact with other particles field
* different ways to conjecture:
-- the individuals are each unique , unlike the particles families in the physical metaphor or source of the analog we're trying to construct. conversely,
-- not unlike individual particle instances of a given class of particles, the invidivual's are identical - eg, "a single soul", and a "single Self" - and thus conform in their various properties, particularly their gauge charge. this is even more attractive than the first conjecture

*
counterconjecture: not unlike physical particles, individs represent instances of different classes / families of particles (having nothing to do necessarilty with biological kinship)
*
each individual generates a field of influence in its neighborhood
*
others are either outside the field or inside, or the field could be infinite tapring off to infinitesimality in the case of people who do not know the individual and who might subissent a residual influence that is negligible not , felt.
*
those that are closer to the indvid who generates his field, have a larger charge wrt to the charge field generated by the individ than others.
*
this determines their proximity or distance to him in his field - which socially translates to the group of people interacting directly with the individ.

*
what kind of topological space would correspond to the set of individuals in a society?
o
how connected, compact, and separable is that topological space?
o
what kind as in how is it to be classified? t0, t1, t2, t2.5, t3, t3.5 , t4, t5, t6,
+
hausdorff, normal, regular, tychonoff, etc.
o
we note that the intersections of neighborhoods are not empty, as one clue to the type of topology of the social spc

*
we let state be the open/closed / clopen sets that constitute the (local) neighborhood of each individ in the social set or space
o
there may be a prblm w/ that def

*
we image that transformations are changes of state for a given subset S of the spc
o
if the change of state is generated from S (self-generated) ,
+
the xform is an automorphism?
+
equiv to action by the subset?
o
if the change is generated by another subset R , it is a xform R→S or is it?
o
it is not
o
we cannot change a subset R to subset S , we cannot transorm people into other people
o
the changes are from state(t1) to state(t2),
o
thus it seems the change is from a subset to a subset hence always an automorphism
o
on the other hand individs may move from one subset to another in the course of / due to a xformat ,thus
+
was the xform applied to single subset ending up affecting more than just that subst or
+
was the xformat applied to all the subsets that got affected?

*
thus there may be two prblms w/ xformats
o
in defining what is a state
o
in defining what is a transformation (of what, what changes what remains the same, from what to what,automorph or not)

*
motivation for social xforms:
o
ideally that would be to look at em in terms of groups, the study of which properties are useful to descriptions (and verifiable sims) of the system
o
but since a) i know neither what it means to organize xforms into groups , nor what the props o groups and their representations, their lie algs and their lie algebras' reps say about the system or the xformats; and b) i am in the effort to understand and know those things
o
this is what the construct of the social field analogy and identifying xformats their groups vis a vis the props / observables and the system.
o
wrk symmetry into that: those xforms that leaves certain props unchanged.


foregoing can thus also relate a bit to an earlier post, I am negative mass.
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Classifying knowledge by energy level

22.v.1430

The various domains of knowledge could be expressed as the functional domain for some knowledge measure function.

by way of example, the study of molecular biological systems that involve low levels of energy would be ranked lower than the study of the fluid dynamics of a waterfall, where the energies measured are much higher. And so the molecular biology would be to the left of waterfall dynamics on an axis or scale of energy levels.

The study of cosmology would probably one of the highest on the knowledge domain axis; and ironically for science, and by definition, theology would occupy the position on the axis with the highest energy level of all - the existence that transcends and contains the natural world and beyond. ad hoc schemes/heuristics to determine the energy level of a given knowledge domain seem feasible.

The initial motivation was to play around with the idea of an integrable knowledge mesaure - eg, a knowledge area function. This presented a challenge which was to find both a measure of knowledge and a metric (to define the domain and distance on the domain of that knowledge measure function, or knowledge area function).

some discussion is at knowledge_area_function.


... Read more

object graph update

02.iI.1430

cramming more stuff in.




excludes properties and morphisms nodes.
if iamge is badley resized try this link ... Read more

note on theorem graphs

02.iI.1430

note on theorem graphs. relating toprevious ones

      * As a possible ideal: theorem graph at the deepest zoom level should include the complete set of cross references in the body of science (ie scientific literature) - though this is also a property of historical math graphs.
        * thus providing on demand sequence/chain paths of book/paper citations, as well as paths of sequences of theorems
      * apologetics:
        * is this not already available in existing literature/existing forms, such as journal indices, encyclopedias, and books?
        * yes but the graph differs in a few things
          * we strip all the descriptive text and other extraneous text and information
          * the whole is presented graphically (and strucurally for machines and report designs)
          * the connections are made explicit and more inclusive and wide ranging than in texts enabling quick cross topic navigation by eye rather than by following links or flipping pages
            * though a counteragrument is that one would still have to pan and zoom to move among paths of the graph

... Read more

social fields?

02.II.1430

* we imagine a human society where each individual is a generating a field, or is the gauge particle of a field.

  * we imagine a human society where each individual is a generating a field, or is the gauge particle of a field
  * or that every individ is endowed with different kinds of charges. 
  * or rather that every individ is both a gauge particle generating a field, and endowed with different charges , "observables that interact with other particles field
  * conjecture: the individuals are each unique  , unlike the particles families in the physical metaphor or source of the analog we're trying to construct
  * counterconjecture: not unlike physical particles, individs represent instances of different classes / families of particles (having nothing to do necessarilty with biological kinship)
  * each individual generates a field of influence in its neighborhood 
  * others are either outside the field or inside, or the field could be infinite tapring off to infinitesimality in the case of people who do not know the individual and who might subissent a residual influence that is negligible not , felt.
  * those that are closer to the indvid who generates his field, have a larger charge wrt to the charge field generated by the individ than others. 
  * this determines their proximity or distance to him in his field - which socially translates to the group of people interacting directly with the individ.

  * what kind of topological space would correspond to the set of individuals in a society?
    * how connected, compact, and separable is that topological space?
    * what kind as in how is it to be classified? t0, t1, t2, t2.5, t3, t3.5 , t4, t5, t6, 
      * hausdorff, normal, regular, tychonoff, etc. 
    * we note that the intersections of neighborhoods are not empty, as one clue to the type of topology of the social spc

  * we let state be the open/closed / clopen sets that constitute the (local) neighborhood of each individ in the social set or space
    * there may be a prblm w/ that def

  * we image that transformations are changes of state for a given subset S of the spc
    * if the change of state is generated from S (self-generated) , 
      * the xform is an automorphism?
      * equiv to action by the subset?
    * if the change is generated by another subset R , it is a xform R->S or is it?
    * it is not
    * we cannot change a subset R to subset S , we cannot transorm people into other people
    * the changes are from state(t1) to state(t2), 
    * thus it seems the change is from a subset to a subset hence always an automorphism
    * on the other hand individs may move from one subset to another in the course of / due to a xformat ,thus 
      * was the xform applied to single subset ending up affecting more than just that subst or
      * was the xformat applied to all the subsets that got affected?
 
  * thus there may be two prblms w/ xformats
    * in defining what is a state 
    * in defining what is a transformation (of what, what changes what remains the same, from what to what,automorph or not)

   * motivation for social xforms:
     * ideally that would be to look at em in terms of groups, the study of which properties are useful to descriptions (and verifiable sims) of the system 
     * but since a) i know neither what it means to organize xforms into groups , nor what the props o groups and their representations, their lie algs and their lie algebras' reps say about the system or the xformats; and b) i am in the effort to understand and know those things
     * this is what the  construct of the social field analogy and identifying xformats their groups vis a vis the props / observables and the system.
     * wrk symmetry into that: those xforms that leaves certain props unchanged.

... Read more

the great natural compressors

the great natural compressors are mathematical formalisms that serve to describe a generalized physical perspective or theory.

For example, there is the group theoretic description of different particles of physics that encapsulates and catalogs their different properties. It is not the only possible way to describe and classify the fundamental particles or fields of nature, but one that happens to be for the moment convenient and popular.

There is also the Hamiltonian formalism to describe the motion of a rigid body , or a chain of them. A system of differential equations can be used to describe completely the thermodynamic behavior and state of an arbitrary physical system. And so on.

The phrase "mathematical formalism" thus refers not merely to a bunch of theorems, but to a distinct coherent scheme of description or model; often one among other possible ones. They encapsulate a great deal of information in terms of two important things. Namely:

(A) the laws and theorems themselves that go into the description or the model.
The formalism usually boils down to (ie, is expressed as / takes the form of a formula of) a single quantity or object
( eg,
the Hamiltonian is the value of the total energy of a system;
the Lagrangian is the action of some segment of a time dependent path (of motion / or in configuration space) defined in terms of some local property of the path ;
the group description is a "representation" (think of it as a set function) of groups of invariant/symmetric and skew/hermitean transformations on observables
).
Despite this, the laws of physics that enter into play in the system being modeled are all derivable from the formalism. For instance, the Lorenz-Maxwell equations of electro-magnetism are all derivable from the Lagrangian formulation. (see [Shankar] ch. 2 for an explicit example of how ; I don't yet know how particle field equations are derivable from group representations).

(B) The formalism readily gives "solutions" that are empirically verifiable when input data is plugged into it.

This is why mathematical formalisms in physics (as in other fields) appear like compression dictionaries (or they are compression schemes). Both functions served by the formalism seem like obvious acts of compression, giving rather fine-detailed and formal descriptions of nature (or whatever the domain of study which is also ultimately a natural subset). These formalisms are a number of conceptual layers up from the underlying "physical theory" (ie the set of proven laws).

They represent a creative act in physical conceptualization / or theoretical thinking, not merely a deductive act. For useful though they are they are interchangeable , not unique. One gaining favor over another by accomplishing a better feat of compression (what is often refered as simplicity, brevity, conciseness, and gaining more generality by separating the field of application from the formal description and manipulation - very much the same way we like to separate content and presentation on computers).

mathematics - ؟ -

These formalisms are even more conceptual layers up from the basic alphabets of literals and idioms and formal patterns we call mathematics. For this seems to be if not the most immediate , than a reasonably good definition of mathematics. It is a set of conventions of linguistic manipulations. Again the spectre of an isomorhpism can be raised between the process of formal reasoning and mathematical expression.

That those conventions have as their by-product the ability to compute, to induct, deduce, the ability to calculate - ie the aspect of calcul or calculus or حساب in mathematics - is the reason we employ them. It's what's great about them.

The quaint thing is, formal though they are, they must be considered part of natural language.


natural vs. formal languages

I do not know why this notion is often ignored (or maybe it isn't) but whatever formal dialects and languages we concoct that are utterly human-readable are to me reasonably considered a subset of natural languages. For two reasons:
1. "we concoct"ed them
2. they are utterly "human-readable".

The arbitrary languages we figuratively feed into state machines in automata theory classes are reasonably formal. Those are the sets of strings acceptable to or generated by FSA ; such as for instance AB*A.

Likewise the programming languages that people develop which are utterly defined , geared to the constraints of a computer architecture are reasonably formal - but frankly the higher-level they are , the more human-readable they become.

But again computer architectures are natural systems.

nature

When we look at what differences there are between the myriad ecological systems that have developed through this planet's history, and the systems that we humans have built ("artificially"), one is tempted to view as a major difference the notion that our artificial systems were consciously developed - rather than emerged.

But nature does not care for conscious acts. although we can study it, bend it to our will, damage it , and hopefully fix it, nature is oblivious to our "will" and our "conscious" efforts.

Our artificial systems are artificial only to ourselves. To any reasonably distant perspective they remain natural systems, as natural as the next quark pair or the next asteroid.

Even within the bowels of human design and architecture, emergence is a common feature and patterns and architectures grow and develop in spite of us.

References

... ... Read more

complex vector algebra must be taught in pre-college curricula

Complex vector algebra (with the generalization of dot product, the inner or scalar product) should be taught in preparatory school curricula , following vector algebra.

This is addition to other absolute necessities in science such as affine geometry, topology , group theory (feasibly one step up from set theory which also are already in pre-college curricula), conditional probability and so on.

It is true not everyone will set out to work in the theoretical physics of fields. Significant patterns in information processing however involve analogues of physical measures (i don't know why they don't call them metaphors) taken in abstract spaces such as data spaces, search spaces, associated weight spaces, etc.

So thus far, complex vector algebra is seen in first in modern physics then in informatics with the myriad applications of its methods in nearly every field.

Sociological research for instance needs to resort to population sampling methods less and less as more comprehensive data is generated state and citizens (societies).

The more creative a researcher can get with what information to induct from and what to search for in the data would depend on how well they can study the input data space. Even though such skills are largely taken up by the software tools they use, the emphasis is on "creative", as in what more information is hidden and not revealed by the classical battery of statistical tools.

In any case, whether all pupils grow up to enter science or not, we all learned real vector algebra in preparatory and secondary grades, and the same should be affored for complex algebra and topology.

So why in prep. grades rather than freshman year at college? Because as is known, the earlier the intake of a technical dialect the more solid its foundation becomes. ... Read more

note on notation and choice of indices

examples of Poisson-Bracket notation: 
  *  [(d)]
  *  [(f)]
  * note how version (f) opts for supercripted indices for one set of canonical
  coords. and subscripted for the other, while the (d) equivalent chooses 
  the simpler notation with all indices subscripted.
  * In [Siegel] [Srednicki] [Aitchison] and others there are discussions on the 
  importance of the choice of indexing scheme used in the QFT commutator formalism 
  - which is already so complicated (;;) .
  * Notation in another version (r) uses square brackets for the PB on the LHS,
  eg. [f,g] , thus making it indistinguishable from the notation used to designate
  the related quantum commutator, the Lie Bracket [a,b]. (Cf. introduction of the 
  commutator in [Shankar])
  * Indeed in __Fields__, Siegel devotes §§A,B in the Symmetry chapter to 
  "Coordinates" and "Indices" resp. [Siegel]
  * [Sussman] calls the superscript indices "traditional"
  * Indexing is usually (almost universally) zero-based. 


-- 24.xi.2008

refs: (of the better/more detailed discussions on indices)

[Aitchison] Gauge theories in particle physics volume 1, 3rd ed.
[Shankar] R. Shankar, Principles of quantum mechanics, 2ed., Yale UP 1992/4.
[Siegel] Warren Siegel, Fields.
[Srednicki] Srednicki, Quantum Field Theory, (c) 2006 , \\ [[http://www.physics.ucsb.edu/~mark/qft.html|book website]]
[Sussman] [Sussman 2001] Gerald Jay Sussman and Jack Wisdom with Meinhard E. Mayer, Structure and Interpretation of Classical Mechanics, The MIT Press Cambridge, Massachusetts, 2001,
http://www-swiss.ai.mit.edu/~gjs/6946/sicm-html/book.html , retrieved 08 aug 2006, 20 nov 2007 and nov/dec 2008. ... Read more

linearity ≅ whole=∑parts?

In general, the main condition or step in proving a set to be a linear space is to show that for any two members f, g of the set under consideration, and for any two reals α,β

a. the linear combination αf + βg is in the set
b. this satisfies closure, from whence follow the remaining axioms 
  if applicable.

As an illusration, consider an operator qcq , L: D → ℝ^ℝ , where D is some set of functions , and ℝ^ℝ is the set of all real valued functions of a real variable.

L is linear, ie, is a linear space if , ∀y∈D, ∀z∈D, ∀α∈ℝ , ∀β∈ℝ

L(αy+βz)= αL(y)+βL(z)

This suggests some meaning. Viz., that operations on / properties of the whole are equal to sums of operations on / props of the parts.

This property appears to be akin to that of self-similarity, suggesting a rather profound meaning for the character of natural organization.

The ubiquity of linearity in the abstract algebra by which we represent natural systems^** (as noted earlier) has such consequences as the integrable character of physical law (discussed elsewhere).

^** if theory were complete then one could say up to an isomorphism, ie, the mathematical formalism is in a one-to-one, onto and domain-covering image of the fields of the physical system being described. ... Read more

baby group

Part of the skill of the craftsmen behind familiar patterns such as this one snipped from Schiller's sprawling FREE book Motion Mountain,





is that the pattern of interwoven (wicker-like) white lines hides the much simpler structure of the underlying patches of color that have well defined shapes. But already by reducing the size of the figure, the color regions become clearer, and so does the simpler underlying design.


To facilitate identifying the color patches , we simplify the pattern by removing the white lines (roughly) and exposing the pattern that was hidden under their clutter.

The colored patches now become clear:





Now we can consider the transformations on this figure that Schiller talks about, and which were unclear before.

The following is largely a distillation of Schiller's discussion .

One of the symmetry transformations that leave the pattern unchanged is rotations in its plane about the origin.

The pattern is unchanged under rotations of pi/2.


It has only four positions in which it appears identical.

In general transformations on a square that leave it looking the same form a group, called dihedral.

Any transformation or bunch of transformations that leave an object appearing unchanged is called a symmetry.

Any collection of symmetries forms a group, a symmetry group.

For a square we can enumerate the possible such symmtery transformations:

rotations and reflections (flips).
We can rotate a square by &plusminus; 90 180 and 270 degrees , leaving it unchanged.
We reflect a square about either diagonal (flip it diagonally)
or reflect the sq. horizontally or vertically (flip it hor. or vertically) and it remains unchanged.

Along with counting the rotation by 0 degree, which is the equivalent of no rotation, which is called an Identity transformation (meaning no-op or do-nothing transformation) , we thus have 8 possible transformations.

To repeat, These form a group called dihedral, hence Schiller refers to it by its symbol as group D₄.

Though Schiller says there is only one transformation - rotation about pi - I don't see it . What i see is that
we have rotations about pi/2 not pi. in other words , yes the figure looks the same when you rotate it 180 degrees,
but it will also look the same when you rotate it ninety degrees. So here either I misunderstand Schiller (more likely)
or something gives.

Apparently he's considering a reflection about pi. yet his group representations use cos n pi/2 and sin n p/2 .

also weirdly he cites a reflection matrix rather than a rotation matrix, unless on (p. 202) he uses a rotation matrix.


With every rotation transformation applied to the figure,
we can see that there are sets of shapes that get transformed into each other with every transformation.

Each set of these identical shapes that get transformed into one another with each transformation is called a multiplet.

To see this, we have the same figure with the multiplet sets enumerated. Again these are the sets of similarly shaped objects that transform into each other under all symmetry transformations (those that leave the properties of the object invariant - here it is the pattern, or the shape - ) such as rotation.

the numbered areas belong each to a multiplet indicated by the number.

Some of the multiplet sets are numbered, with the elements of each set given the number of that set.

Some multiplets are not numbered, but have arrows pointing at them.

Each multiplet as a whole (its entire set of elements) has the same symmetry as the overall figure.

For some of the colored shapes, the multiplet needs four objects to make up a whole multiplet (e.g.,
in the case of multiplet numbered 1 , we have identified its four elements).

Another multiplet, number 5, looks like it has 8 elements, but I am guessing since the symmetry degree of the multiplets is the same as that of the group.

Schiller points out another multiplet that has only one member, the central star. It is a one-element multiplet.

In any symmetry system such as this arabesque pattern, each part or component of the system is "classified" by the type of multiplet it belongs to.

Multiplets are also known as group representations.

More formally than the kindergarten chitchat above,

the representation of a group (aka symmetry group) is

an assignment of linear transformation matrix A(g) to each group element s.t.
A(g ∘ h) = A(g) ∘ A(h)

More formally still it is a homomorphism from the space of linear transformations onto the elements of the symmetry group.

Representations of unitary transformations are called unitary. Unitary transformations are matrices s.t. A^*=A^-1.

These have eigenvalues of norm (value?) 1 (perhaps leading to principle of gauge invariance? - afterall a gauge is a seminorm, and since norm is always one, gauge is invariant?) and mappings of unitary matrix are one-to-one.


If a matrix is not 121 it is singular, its determinant is zero, and it has no inverse transformation.

Almost all representations appearing in physics are unitary.

time evolution of physical systems is always described by the always one-to-one unitary mappings because these
map from time t-1 to t in a one-to-one fashion.

All unitary representations correspond to transformations that are one-to-one and invertible (both properties ,
equivalent to saying that determinant of the matrix is not zero and the matrix is thus non-singular.

A matrix whose det is zero is called singular as it has no inverse matrix. It cannot thus belong to a group.

Schiller then discusses reducibility of representations , identifying submultiplets , and giving rise to a classification of representations.

E.g., the pattern's symmetry group (called approximate) "has eight elements.

It has the general faithful unitary and irreducible group representations." and is an octet

Giving rise to the notation D₄(The so-called "approximate symmetry group") in the given figure, denoted D_4 , has eight elemnts.)

In any case it is from the symmetry that the deduction of the "list of multiplets or representations" that describe the symmetry's "building blocks" is possible.

Apparently other symmetry groups are given for other multipets? ie the representations are reducible to singlets doublets and quartets? This is what Schiller seems to say.

Schiller notes how unlike the transformations of the tiling pattern , which were discrete, the viewpoint transformations under which the world demeures unchanged are continuous and unbounded.

Siegel's Fields also states that continuous symmetry "is one of the most fundamental and important concepts of physics."

Continuity of the xforms means their representations are continually variable, without bounds and notably are magnitudes. In other words, scalars - as opposed to vectors. They can only be scalars.

Schiller notes ominously that in contrast vectors and tensors "only scalars may take discrete values", "may be discrete observables." (p. 204)

But weren't representations just made continually variable b/c nature's symmetry transformations are continuous? sounds a bit coucou .

To make things worse Schiller states that most representations also possess direction.
So they're not scalars anymore ?
Also it is states that symmetry under change of observ. pos inst or orientation => all observables are scalars, vectors, tensors or spinors (in asc. ord. of generality).









BTW rotations form (i.e., are) an Abelian group because g ∘ h = h ∘ g.

And generally it seems that a group has higher symmetry than its subgroups; iow, a group is a "larger symmetry group" than its subgroups. Makeosa della sensa.



* * *


The b&w lines thrown on top of the layer of multiplets (the color patches or geometrically shaped regions) (that get mapped to each other when undergoing transformations like rotation) are not mere spaghetti thrown on top of the rotation group representations.

That is to say, the layer of interlocking b&w lines is not trivial. the new layer is describable - i guess - by a greater number of representations for the transformation group of this pattern. (not sure this is correct though)

... Read more

no makedy sensa de gauge invariance: call this a narrative?

[ note: at present this note concatenates these two posts:
more-notes-for-narrative,
ideas-for-invariance-narrative;
which isolate notes from quatum_mechanics and gauge_theory - ]

The scale of the sarh, the building that is modern physical formalism is staggering. I almost cried (i wish i did). if I had been able to better understand I may have shed tears. The quantum field theory formalism is a vast structure of theoretic thought with the complexity of a very large engineering
feat (though it is a description of nature, what appears like a singular engineering feat, except that the term "engineering" would not correctly apply in this case).

It is fine-detailed , and yet one by one the steps in developing / deriving the formulations are simple and elegant but also speak of mathematical genius , command and audacity , although truth be told, proofs are rare in qft.

The indirection in terms of higher-level abstractions of lower-level primitives is manifold.

It is probably the most indirection i've had to deal with wrt any given subject ?

We (rather they, the specialists) are talking about structures of mappings or homomorphisms on groups of continuous unitary transformations (or xform space)
that (also are invariant with respect to their norm as they are unitary ? and ) leave invariant groups of scalar observables (measurable variables, dependent on particles and or time) which are derivable from (in a rather two-way fashion) differential field equations.

In terms of symmetry descriptions of nature are legion and eloquent , thus deserving their own note.

the quantized schrodinger wave equation in the Hamiltonian and state function

has complex solutions. These form a unitary complex vector space with a (complete?) metric (think distance, norm , seminorm, gauge (?)) that has unitary linear self-adjoint operators

much like there are dual spaces in the case of a linear space's inner product
(also think the dual orthogonal spaces of the physical em field ) .

... and so on ...

Though the narrative is neither dense nor tight nor clear yet,
there's enough coalescence to hold one's hat on for the next bout of reading.

All this with a recording of Ramsey Lewis' cover of "That's the way of the world" playing.
This along with a momentous playlist that's been very smooth. Then again, which of my playlists have not been smooth? Well, it did arise that tracks that were dissonant with playlist's general mood push a number of times me to wonder "how did that get on here?" , particularly when trying out the playlists to entertain others. But insofar as a study score, this list is doing fine.
A listing of some of the tracks can be seen in the sidebar on the right .

• 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 )
  
  
  
  
  
  • 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.)
  
  

... Read more

linear combos everywhere

• a linear combination is a finite sum of the form
  
  • [Shankar] calls it a superposition. ([Shankar] §1.2 p. 9).
    
    
    
    • there is also something called the superposition principle:
      
      
      
      • F(x)=A,F(y)=B ⇒ F(x+y)=A+B
        
        
      
      
      • additivity property of functions making them linear functions,aka lin. operators and lin. maps.
        
        
      
      
      
      
    
    
    
    
  
  
  • a finite sum of that form
    
    ⇔ summation of terms
    
    ⇔ sum of products
    
    
  
  
  
  


• One clear physical meaning of the linear combination is that it is a weighted sum.
  

• instances of linear combination:
  
  
  
  
  
  • linear combinations are thus a key primitive in - mathematically speaking - "nature".
    
    
  
  
  • instances of linear combination in inner products or related by the inner product:
    
    
    
    
    • inner product is a class of operations having the same set of properties,e.g.,
      
      
    
    
    
    • vector dot product
      
      
      
      
      • of two vectors A⋅B=Σaibi , and in its variant, matrix multiplication , thus
        
        
      
      
      
      
    
    
    • matrix multiplication
      
      
      
      
      • , for each row in the mxn premultiplier elements from each column in the row are linearly combined (linearly combine) with elements from the corresponding row element in the postmultiplier giving a single element in the product matrix
        
        
        
      
      
      • cij = ai1b1j + ai2b2j + ai3b3j + … + ainbnj
        
        
      
      
      
      
    
    
    • integration operator
      
      
      
      
      • 1.bis2- in the approximation of integration by finite sums
        
        
      
      
      
      • Σf(ci)Δx , where ci is a point in the ith subinterval
        
        
      
      
      • the integral itself expressing a wide class of functions eg, area,distance,path length,volume,work,flow,etc. = ∫f=limΣf(x)⋅(Δx→0)
        
        
      
      
      • n.b. the integrat operator is a linear transformation, which have linear combination as a property, see below.
        
        
      
      
      
      
    
    
    • N.B. N.B. the equivalence of integrals and the dot product. Explicit examples:
      
      
      
      
      
      • [Waerden]§1: where to express that state functions Ψ are Lebesgue-integrable, he writes <Ψ,Ψ> = ∫ΨΨ*dq, where I take the LHS to be an inner product.
        
        
      
      
      • in investigating, we can take this parallelism further by emphasizing the coef:
        
        
        
        
        • case: dot product, matrix multiplication and determinant: coefs is a "vector" component
          
          
          
        
        
        • case: finite sum: coef. is subinterval width, Δx=b-a/n and kth x is x_k=x0+k(b-a/n)
          
          
        
        
        
        
      
      
      
      
    
    
    • The Unit form , Hermitean product <u,v>
      
      
      
      
      
      • the unit form <v,v> = ∑ c_k^* c_k ([Waerden] §9)
        
        
      
      
      • is pretty damn close to the way
        
        
        
        
        • state functions Ψn of the Schrodinger equation are "integrable in the sense of lebesgue" , ie <Ψ,Ψ>=∫Ψ^*Ψ ([Waerden] §1)
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  • Determinant calculation , for orders 2 and 3 at least ?
    
    
  
  
  • the definition of a vector itself. As a vector is expressible as a linear combination of the magnitudes of its components and the orthonormal basis vectors, better known to us as unit coordinate vectors.
    
    
    
    
    • eg, a vector in Euclidian space may be expressed as
      
      
      V = v_x(1,0,0)+v_y(0,1,0)+v_z(0,0,1) = +v_x.i++v_y.j+v_z.k .
      
      see theorem 12.6 in [Aposotl I].
      
      
    
    
    • a vector in vector space of n-tuples V_n is expressed as a linear combination of the space's unit coordinate vectors (eg, i,j,k for V₃).
      
      
      
      
      
      • [ApostolI]ch15,§15.6: definition: the set of all fin. linear comb. of elmts of linear space S
        
        
        
        
        • satisfies closure,
          
          
        
        
        • is a subspc of S and
          
          
        
        
        • is called the span of S, or subspc spanned by S.
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  • Linear transformation: application of Linear transformation A to finite-dimensional vector space V:
    
    
    
    
    • lin. xforms are matrix multiplications, so by extension of this and by definition ipso facto are also linear combinations
      
      
    
    
    • 
      
      
      
    
    
    • !@ see third property of linear transformations in [Apostol II] 2.1. !@
      
      
    
    
    
    
  
  
  • Gram-schmidt process of orthogonalization
    
    
    
    
    • (mapping among either basis sets or axes or both)
      
      
      
    
    
    • [Apostol II] pp.24-26.
      
      
    
    
    
    
  
  
  
  

• The definition of a linear manifold [von Neumann] §II.1 p. 38.




• "a subset U of a linear space R is called a linear manifold if it contains all linear combinations
  
  a₁f₁ + … + a_kf_k for any k of elements f₁, ... , f_k.




• "[sufficiently requiring]" ( f,g ∈ R ) ⇒ ( af, f+g ∈ R )




• .'. f₁,…,f_k ∈ U ⇒ a₁f₁ , a₁f₁+a₂f₂, a₁f₁+ a₂f₂+a₃f₃, ..., a₁f₁+...+a_kf_k ∈ U


• or, put slightly differently,


• .'. ∀ f₁,...,f_k ∈ U   a₁f₁ , a₁f₁+ a₂f₂, a₁f₁+ a₂f₂+a₃f₃, …, a₁f₁+...+a_kf_k ∈ U.






• U is ⊆ R ⇒ the set a₁f₁ + … + a_kf_k ∀ k=1,2,…, a₁,…,a_k in ℂ, f₁, …, f_k in U , it is a subset of every other linear manifold , which is then said to be spanned by U.

... Read more

group terms - interim

Source code can be seen by clicking on [pagesource] at
http://augmented5th.dyndns.org/doku/doku.php?id=math_object_graphs ... Read more

groups with maps - interim

 




Dot script Source code can be seen by clicking on [pagesource] at bottom of
http://augmented5th.dyndns.org/doku/doku.php?id=math_object_graphs

This is one of the more disastrous graphs because for now both object and operation clusters are side to side putting most links between the two node sets into a bundle.

Hopefully that will change later with more objects added. ... Read more

note regarding group theory

... Read more

mathematical prerequisites for gauge theories

 


where yellow box = theory , and green ellipses=objects.

Information in the above is from the two texts,

George Svetlichny, "Preparation for gauge theory", arXiv:math-ph/9902027v3 12 Mar 1999. (a quick review).

Aitchison & Hey, Gauge Theories in Particle Physics, Volume 1: From Relativistic Quantum Mechanics to QED, Third Edition, CRC Press, 2002.

From a theoretic-organization perspective, a different graph is needed, an abridged draft of which is shown here,



p.s. From a graphing perspective, the top graph contains the information content of only about a minute's worth of reading. Which unfortunately does not bode very well for the general utility of theorem graphs considered in an earlier note. ... Read more

Theorem graphs

Theorem graphs are as useful as text in conveying information on the mathematical or theoretical subject being studied.

There are several types of graphs or graph categories and an application is needed to let users manipulate those graph category to cover a particular theorem , theory or field of study.

Such an application should enable users to select what types of graphs they want to view or update (edit)

as well as to view more types of graphs at the same time

The ability to zoom in and out navigate edit and update the graph

Zooming should be implemented both
a) visually (the familiar zoom in out scaling)
b) and structurally by the ability to "collapse" a group of nodes or subgraph (if the subgraph is collapsible) into a single node and to expand a node to a collection of nodes or subgraph , if the node happens to be "expandable".

This enables the user's navigation across layers of abstraction or layers of detail, and lower level of the theoretic body.

The graphs should cover a wide range of fine-ness , from the most general, broad label, type graphs ,

to the very fine structure of the elements that go into a definition or a theorem.

In the limit, a theorem graph should be able to "graphize" even the elements of a proof of a theorem, through a combination of logic and object graphs.

The Types of graphs:

object graphs: where
nodes tend to represent
- entities , such as sets and sequences
- operations, such as binary relations or maps (eg, functions)
- properties, such as linearity, commutativity
edges to to represent a relation of the type "this goes into this, or makes up this"
more formally, the child "has" the parent property or relation or set

logic graphs
where emphasis is on meanings of edges, which include:
parent proves child
the edges represent a sequence of steps in a proof, with the ultimate node being
an enunciation of the theorem
There is still some ambiguity on the choice of what nodes are to represent.
They can represent steps in a single proof , in sequence ,
where each step is node


historical graphs
nodes include
writings
persons
experiments
date node: plaintext nodes with "year" labels connected by invisible edges to make a timeline, as seen in the old unix
history dot graph.

Nodes also include
definitions
theorems
hypotheses , conjectures, etc.
theories : more general than theorem, and generally correspond to a distinct field of study as well as part of a theoretic framework

Edges in historical graphs often denote "contributes to" as well "leads to"

Because there are different categories of graphs, the user should be able to switch graph categories on and off according to their convenience.

In a historical graph, a user should be able to click on a node for a given theorem, and thus expands a subgraph which in fuller detail, shows the user depending on his choice, either the object or logic makeup of the theorem or definition.

Examples with static graphs and more notes can be found at math_graphs. ... Read more