Polyphyly Updated 2025-07-16
Basically mean that parallel evolution happened. Some cool ones:
Primate Updated 2025-07-16
TODO holy crap, even this is hard to understand/find a clear definition of.
The Dirac equation, OK, is a partial differential equation, so we can easily understand its definition with basic calculus. We may not be able to solve it efficiently, but at least we understand it.
The formulation of QFT also appears to be a form of infinite-dimentional calculus.
Quantum electrodynamics by Lifshitz et al. 2nd edition (1982) chapter 1. "The uncertainty principle in the relativistic case" contains an interesting idea:
The foregoing discussion suggests that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta,
polarizations) of free particles: the initial particles which come into interaction, and the final particles which result from the process.
Projective elliptic geometry Updated 2025-07-16
Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.
Projective space Updated 2025-07-16
A unique projective space can be defined for any vector space.
The projective space associated with a given vector space is denoted .
The definition is to take the vector space, remove the zero element, and identify all elements that lie on the same line, i.e.
The most important initial example to study is the real projective plane.
Endocytosis Updated 2025-07-16
Mathematician Updated 2025-07-16
Poet, scientists and warriors all in one? Conquerors of the useless.
A wise teacher from University of São Paulo once told the class Ciro Santilli attended an anecdote about his life:
I used to want to learn Mathematics.
But it was very hard.
So in the end, I became an engineer, and found an engineering solution to the problem, and married a Mathematician instead.
It turned out that, about 10 years later, Ciro ended up following this advice, unwittingly.
Figure 1.
xkcd 435: Fields arranged by purity
. Source.
Matrix mechanics Updated 2025-07-16
It is apparently more closely related to the ladder operator method, which is a more algebraic than the more analytical Schrödinger equation.
It appears that this formulation makes the importance of the Poisson bracket clear, and explains why physicists are so obsessed with talking about position and momentum space. This point of view also apparently makes it clearer that quantum mechanics can be seen as a generalization of classical mechanics through the Hamiltonian.
Inward Bound by Abraham Pais (1988) chapter 12 "Quantum mechanics, an essay" part (c) "A chronology" has some ultra brief, but worthwhile mentions of matrix mechanics and the commutator.
As usual, it is useful to think about how a bilinear form looks like in terms of vectors and matrices.
Unlike a linear form, which was a vector, because it has two inputs, the bilinear form is represented by a matrix which encodes the value for each possible pair of basis vectors.
In terms of that matrix, the form is then given by:
Maxima and minima Updated 2025-07-16
Given a function :
we want to find the points of the domain of where the value of is smaller (for minima, or larger for maxima) than all other points in some neighbourhood of .
In the case of Functionals, this problem is treated under the theory of the calculus of variations.

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