In mathematics, physics and other disciplines a theorem or similar statement is sometimes said to be folklore when the statement was accepted by the community without a recognized proof:
A complete orthonormal system of eigenstates of an observable (e.g., the energy operator) cannot be extended to a larger orthonormal system of eigenstates [2]
Additive groups play a crucial role in algebraic topology. For example, homology and cohomology groups are used in order to introduce Betti numbers and torsion coefficients as fundamental topological invariants. [3]
Algebraic deformation represents a method for the quantization of Lie groups and Lie algebras. Quantum groups are not groups, but Hopf algebras. [3]
An important task of physics is to compute effective physical constants which average microphysical effects on a macroscopic scale. [1]
Banach spaces are used in order to study the convergence of iterative methods which play a fundamental role in perturbation theory. [1]
Behind renormalization there lurks a monster called the motivic Galois group. This monster is responsible for the rich mathematical structure of renormalization theory. [1]
Bifurcation is caused by a loss of stability; it is accompanied by an increase of complexity, and hence a decrease of entropy. [1]
Both the covariant and the contravariant transformation law play a key role in classical tensor calculus. [3]
Both the Wightman functions and the correlation functions of the quantized harmonic oscillator are the prototypes of general constructions used in quantum field theory. [2]
Causality is crucial for quantum field theory. [1]
Celestial mechanics, quantum mechanics, and quantum field theory are governed by perturbation theory. [3]
Classical derivatives are generalized to functional derivatives; differentials are linear functionals in modern mathematics. [1]
Classical wave optics culminates in microlocal analysis. [1]
Computation of cross sections means computation of traces for products of Dirac matrices. [2]
Contact transformations provide the general setting for transforming ordinary and partial differential equations in such a way that solutions of the original equation pass over to solutions of the transformed equation. [2]
Cotangent spaces are the dual spaces to tangent spaces. [3]
Distinguish carefully between well-posed and ill-posed problems. [2]
Distinguish strictly between free moments and full moments. [2]
Duality plays a crucial role in the theory of topological groups; characters generalize the exponential function. [3]
Einstein emphasized the importance of invariants in physics. [3]
Elementary particle processes are invariant under the CPT symmetry. This is one of the most fundamental symmetries in physics. [1]
Elementary symmetric polynomials can be replaced by completely symmetric polynomials as generating polynomials for symmetric polynomials. [3]
Eliminate virtual photons as external particles. [2]
Equivalence classes and the corresponding quotient structures (e.g., quotient groups, quotient algebras, quotient fields) appear everywhere in mathematics. [3]
Extend the field R of classical real numbers to the field ∗R of generalized real numbers in order to get infinitesimal numbers and infinite numbers. [2]
Feynman’s approach to quantum theory can be understood best by using Dirac’s formal calculus; this can be generalized straightforward to quantum field theory. [2]
Feynman’s path integrals are both infinite-dimensional Gaussian integrals and continuous partition functions. [1]
Finite groups can equivalently described by incidence numbers. [2]
For a linear operator, the change of the basis of the underlying linear space corresponds to a similarity transformation of the matrices. [3]
For quantum mechanics, it is crucial to replace eigenvectors by eigencostates. [2]
Force equals curvature in modern physics. [1]
Formal manipulations in mathematics can lead to completely wrong results. [2]
Functionals generalize classical functions to systems with a finite or infinite number of degrees of freedom. [1]
Functors play a crucial role in modern mathematics and physics. [2]
Functors reduce the solution of topological problems to simpler algebraic problems. [1]
Generating functions are used in physics in order to encode the properties of multi-particle systems in statistical physics and quantum field theory. [3]
Generating functions for the moments of random variables represent a basic tool in quantum field theory [2]
Groups describe symmetries in physics, whereas the linearization of symmetries leads to Lie algebras. [1]
In a natural way, the global mathematical description of physical fields is based on the language of bundles. [2]
In contrast to the underlying complex Hilbert space, the space of quantum states has a nontrivial topological structure. [2]
In contrast to the Wightman functions, the correlation functions reflect causality. [2]
In formal terms, the ground state energy (vacuum energy) of the electromagnetic quantum field is infinite. This causes mathematical trouble in quantum electrodynamics. [2]
In general topological spaces, classical sequences are not enough for the characterization of closed sets by convergence. [2]
In Hamiltonian mechanics, we pass from position and velocity to position and momentum. [2]
In mathematics and physics, many-particle systems are described by generating functions. Physicists call them partition functions. [1]
In mathematics and physics, one encounters much more semidirect products of groups than direct products. [3]
In mechanics it is important to use the appropriate coordinates. [2]
In modern differential geometry, velocity vector fields, covector fields, and tensor fields on manifolds are described by sections of the tangent bundle, the cotangent bundle, and the tensor bundle, respectively. Later on, this will allow us to study the global (i.e., topological) properties of physical fields. [3]
In modern mathematics, the variations δx, δy and δS are not mystical infinitesimals, but well-defined finite mathematical quantities. [3]
In modern physics, the Huygens principle is replaced by the parallel transport of physical information. [2]
In order to get insight in differential geometry, use differential forms and employ their invariance properties. [3]
In order to understand classical mechanics and quantum mechanics, one has to understand geometrical optics. [2]
In order to understand the beauty of Feynman’s approach to quantum mechanics, one has to understand the Brownian motion of immersed particles and its relation to diffusion processes. [2]
In particle accelerators, many particles are unstable; such so-called resonances only live a very short time [2]
In perturbation theory, the scattering of free quantum particles under the action of a force is described by the Heisenberg scattering operator. The kernel of the scattering operator can be represented by the propagator kernel. This is the second magic formula in quantum physics [2]
In physics, effective quantities can be obtained by averaging. [1]
In quantum field theory, Green’s functions are closely related to distribution-valued meromorphic functions. [2]
In quantum field theory, try to fix gauge potentials by using convenient gauge conditions. [1]
In semiclassical statistical physics, the extended algebra of observables is a commutative ∗-algebra of functions, and the states are generated by some probability measure. [2]
In terms of physics, Artin’s braid groups describe interactions in nature based on a special type of twisting. [3]
In terms of physics, Feynman propagators describe the propagation of physical effects in quantum field theory by taking both causality and antiparticles into account. In terms of mathematics, Feynman propagators are distinguished fundamental solutions of the wave equation, the Klein–Gordon equation, and the Dirac equation. In the Fourier space, Feynman propagators are inverse differential operators, after regularization. [2]
In terms of quantum physics, the Fourier transform relates the position space to the momentum space. [1]
In the perturbative approach to quantum field theory, combinatorial formulas are used for computing correlation functions of interacting quantum fields by means of simpler correlation functions of free fields. [2]
Infinitesimal symmetry transformations know much, but not all about global symmetry transformations. [1]
Integrable systems are only approximations of realistic systems in nature. [2]
Integral transformations are extremely useful in mathematics and physics. [1]
Integration by parts is the key to the calculus of variations and to the modern theory of linear and nonlinear partial differential equations. [1]
Invariance under rotations leads to conservation of angular momentum. [2]
Invariant theory is essentially based on the Leibniz chain rule in differential calculus. [3]
Irreducible representations are the atoms of representations. [3]
Lie groups describe finite symmetries or symmetries which smoothly depend on a finite number of real parameters. Lie algebras are the linearization of Lie groups at the unit element. The passage from Lie groups to Lie algebras simplifies considerably the approach. Lie algebras are frequently called infinitesimal symmetries. [3]
Light is not only the basis of our biological existence, but also an essential source of our knowledge about the physical laws of nature, ranging from the seventeenth century geometrical optics up to the twentieth century theory of general relativity and quantum electrodynamics. [2]
Local functional derivatives are frequently used in quantum field theory. They generalize classical partial derivatives to an infinite number of variables. [1]
Many important manifolds arising in mathematics and physics (e.g., Riemann surfaces) can be equipped with an Hermitean Kaehler metric. [2]
Many time-dependent processes in physics and engineering become simple in the frequency space via the Fourier transform. [1]
Many-particle systems in nature are able to store information. This is equivalent to both the measure of disorder and the notion of entropy in physics. [2]
Mathematicians and physicists enjoy Hamiltonian systems, since such dynamical systems allow the application of methods from symplectic geometry. In particular, the Hamiltonian function of a Hamiltonian system is always a conserved quantity. [2]
Mathematicians and physicists like holomorphic and meromorphic functions, since the local behavior of such functions determines completely their global behavior. [1]
Mnemonically, the Dirac calculus works perfectly. [3]
Mnemonically, the principle of killing indices works on its own. [3]
Multilinear algebra studies all kinds of products. In nature, one observes fusion and splitting of physical states. From the mathematical point of view, this corresponds to products and coproducts of Hopf algebras, respectively. [2]
Nonlinear problems describe interactions in nature. In contrast to nonlinear problems, linear problems enjoy the superposition principle. The advantage of perturbation theory is, that it reduces nonlinear problems to linear problems. Therefore, perturbation theory is widely used in physics. [3]
Non-standard analysis adds infinitesimals and infinite numbers to the classical real line. [2]
One of the main tasks of mathematics and physics is to simplify extremely long computations by getting insight into the symmetry structure of the expressions. [3]
Only low-dimensional algebras possess a nice structure. [3]
Orientation plays a fundamental role in physics. In particular, there are physical processes which depend critically on the choice of orientation. [3]
Overlapping divergences caused a lot of trouble in the history of renormalization theory. [2]
Partially ordered sets occur quite often in mathematics. [2]
Partition functions are the main tool for studying many-particle systems. [1]
Perturbation theory is the most important method in modern physics. [1]
Photons, electrons, and positrons are described by the tensor product of the corresponding indefinite and definite Fock spaces. [2]
Physical fields are sections of fiber bundles. The qualitative (i.e., topological) structure of physical fields is determined by the topological structure of the corresponding fiber bundles. The prototype of a fiber bundle is the tangent bundle of a manifold. [3]
Physical fields can be described by bundles in mathematics. The change of the real values measured by different observers corresponds to cocyles. [2]
Physical states are equivalent iff they lie on the same orbit generated by the action of the gauge group. Therefore, gauge theory has to be based on orbit spaces (also called moduli spaces in mathematics). This complicates substantially the mathematical theory. As a rule, moduli spaces are not manifolds; they possess singularities. Surfaces or more general varieties with singularities are studied in algebraic geometry. [3]
Physically, partitions play a crucial role in the theory of multi-particle systems. The basic idea is to describe the physics of the total system by the physics of all possible subsystems. Mathematically, partitions govern the representation theory of permutation groups. The number of essential irreducible linear representations of a permutation group is equal to the number of orbits of the permutation group with respect to the inner automorphisms; this is equal to the number of conjugacy classes. In turn, this is equal to the number of partitions of the group order. Graphically, this equals the number of Young diagrams. [3]
Physicists measure real numbers in experiments. The theory has to predict these numbers. [2]
Products on real (resp. complex) linear spaces can be reduced to linear operators on tensor products. This way, multilinear algebra can be reduced to linear algebra. [3]
Products play a fundamental role in quantum field theory (e.g., normal products, time-ordered products, retarded products). They are used in order to construct correlation functions. [2]
Propagation of quantum effects differs substantially from classical physics. [2]
Quantization is not a handicraft, but an art. [2]
Quantum electrodynamics couples the Maxwell equation for the photon to the Dirac equation for the electron. [2]
Quantum field theory is based on only a few basic principles which we call magic formulas. [1]
Quantum field theory studies the creation and annihilation of particles. [1]
Quantum fields possess an infinite number of degrees of freedom. However, in order to overcome serious difficulties, it is wise to start with a finite number of degrees of freedom and to study the lattice limit (continuum limit) for such quantities which can be measured in physical experiments. [2]
Quantum fields possess an infinite number of degrees of freedom. This causes a lot of mathematical trouble. [1]
Quantum physics enforces the study of noncommutative mathematical structures. [3]
Quantum physics is based on the study of states, costates, and observables. [1]
Quantum states are equivalence classes. Global physical fields are sections of bundles. [2]
Reduce constrained extremal problems to free extremal problems by changing the Lagrangian. [1]
Regularize divergent integrals by introducing additional ghost particles of large masses. [2]
Relativistic invariance forces the use of indefinite inner products. [2]
Representation theory is governed by the highest weight. [3]
Representations of symmetries with the aid of linear operators (e.g., matrices) play a crucial role in modern physics. In particular, this concerns the linear representations of groups, Lie algebras, and quantum groups (Hopf algebras). [3]
Resonances are dangerous for the mathematics of physical systems. [1]
Rigorous propagator theory is based on von Neumann’s operator calculus for functions of self-adjoint operators. [2]
Scalar photons can never exist in a Hilbert space. [2]
Solvable groups are closely related to commutative groups. [3]
Solvable Lie algebras are close to both upper triangular matrices and commutative Lie algebras. In contrast to this, semisimple Lie algebras are as far as possible from being commutative. By Levi’s decomposition theorem, any Lie algebra is built out of a solvable and a semisimple one. The nontrivial prototype of a solvable Lie algebra is the Heisenberg algebra. [3]
Spherical geometry is the geometry on the surface of earth. [2]
Study the response of physical systems under the influence of external sources. [1]
Symmetric and antisymmetric functions play a fundamental role in mathematics and physics, for example, in representation theory and topology (e.g. the construction of topological invariants like characteristic classes). [3]
Symmetries essentially simplify computations in mathematics and physics. [1]
Symmetries of the action functional lead to symmetries of the Euler– Lagrange equations. In particular, invariance of the action functional under time translations is responsible for the conservation of energy. Degeneracy of the second variation generates local symmetries also called gauge symmetries. The use of symmetries is basic for modern physics. [3]
Symmetrization and antisymmetrization play a crucial role in mathematics and physics (e.g., for constructing invariants). For example, bosons (e.g., photons) are based on symmetrization, whereas fermions (e.g., electrons) are based on antisymmetrization. [3]
Tangent spaces are spaces of velocity vectors. [3]
The action knows all about a quantum system via functional integrals. [1]
The approach to problems in physics can be simplified by passing to an appropriate system of reference which fits best the physical situation. [3]
The big surprise in renormalization theory is the appearance of unexpected huge cancellations in the lengthy computations. [2]
The calculus of variations has its roots in extremal problems for real-valued functions. [2]
The center of gravity moves like a mass point equipped with the total mass under the action of the total force. [3]
The coinverse (also called antipode) of Hopf algebras allows us to elegantly describe complicated inversion processes in mathematics and physics. [2]
The Compton effect lies at the heart of modern physics. [2]
The computation of scattering processes in quantum field theory can be elegantly reduced to the computation of propagators by using the Wick theorem. [2]
The concept of duality is crucial in both mathematics and physics. [1]
The definition of infinite-dimensional Gaussian integrals depends on the spectrum of the linear symmetric dispersion operator. [2]
The development starting with Gauss culminated in Poincaré’s creation of the theory of Fuchsian groups and automorphic functions. [2]
The dynamics of a quantum system is described by a time-dependent operator called the Feynman propagator. The kernel of the propagator can be formally represented by a Feynman path integral which depends on the classical Hamiltonian (i.e., the symbol of the Hamiltonian operator). This is the first magic formula in quantum physics. [2]
The elementary symmetric polynomials are homogeneous polynomials which can be regarded as the atoms of symmetric polynomials. [3]
The exponential function is the most important function in mathematics [2]
The extended quantum action functional knows all about the quantum field. [1]
The family of moments knows all about a given random phenomenon in nature. [1]
The Feynman path integral encodes the correlations (i.e., the Green functions) of a quantum field. The main task is to decode the information. [3]
The finite Monster group brings together modern algebra and conformal quantum field theory. [1]
The flow of ideas from physics to mathematics and vice versa is crucial. [3]
The Fourier transformation and its generalizations lie at the heart of mathematics. [2]
The free Hamiltonian is a paradigm for general Hamiltonians in quantum mechanics and quantum field theory. [2]
The fundamental Levi decomposition of a Lie algebra is the prototype of a semidirect product of Lie algebras. [3]
The Gaussian elimination method is a universal method for solving finite-dimensional linear matrix equations on computers. [3]
The global behavior of the quantized harmonic oscillator is governed by the Morse indices (also called Maslov indices) of the classical harmonic oscillator. [2]
The Green’s function allows us to reduce the solution of general boundary values to the solution of a special boundary value problem. [1]
The history of mathematics shows that every well-working formal calculus used in physics can be rigorously justified once a day, by finding the appropriate rigorous tools. [2]
The Huygens principle is the first general principle in the history of physics, which describes the propagation of physical effects. [2]
The idea of Clifford algebra is basic for Dirac’s theory of the relativistic electron, and hence it is crucial for the fundamental fermions in the Standard Model in particle physics. [3]
The interactions between elementary particles in the Standard Model are described by gauge field theories. [1]
The introduction of additional unphysical states called ghosts helps to obtain a clear presentation of quantized gauge theories. The key point is that, roughly speaking, the physics observed in experiments does not depend on the ghosts. [3]
The Lagrangian trick is to use a variational principle with respect to local coordinates for the manifold of constrained positions generated by the constraints. This way, the constraints drop out. [3]
The Lebesgue integral was the key to modern analysis based on functional analysis in the twentieth century. [3]
The method of highest weight is used by physicists in elementary particle physics because of its simplicity and elegance. [3]
The morphisms of a simple group are trivial. Simple groups can be regarded as the atoms in group theory. [3]
The Morse index describes the global behavior of the action functional of the harmonic oscillator with respect to arbitrary time intervals. This global behavior is governed by the appearance of focal points of the trajectories of the harmonic oscillator, which correspond to focal points in geometric optics. [2]
The most important probability distribution is the Gaussian distribution. [1]
The Moyal star product of classical symbols passes over to the operator product of the corresponding Weyl operators. [2]
The notion of limit for sequences in a topological space can be generalized to the limit of mathematical structures (e.g., linear spaces, groups, topological spaces). [2]
The notion of manifold is of fundamental importance for both mathematics and physics. There arises the problem of classifying manifolds in terms of topology. This has been one of the most important research topics in topology in the last 150 years. [2]
The operator kernel knows all about the operator. [2]
The partition function knows all about the thermodynamic system. The Feynman path integral can be viewed as a generalized partition function. [2]
The path integral for the harmonic oscillator is closely related to the difference method for the classical harmonic oscillator in numerical analysis. [2]
The Poincaré model is the most elegant formulation of hyperbolic non-Euclidean geometry [2]
The Poincaré–Wirtinger calculus reformulates real analysis in terms of the language of complex analysis. This is very useful for modern quantum theory. [2]
The raising operators in combinatorial mathematics can be regarded as simplified models for particle creation and particle annihilation in quantum field theory. [3]
The representation theory for groups is an extremely useful tool in order to understand the structure of both molecule spectra and scattering processes for elementary particles. The same concerns the structure of processes for nuclei of molecules. [1]
The rescaled Fourier transform fits best the duality between position and momentum of quantum particles in the setting of the Dirac calculus. [2]
The S-Matrix knows all about scattering processes and bound states of quantum particles. [2]
The space of quantum states of a quantum system possesses a nontrivial topology. [1]
The spin of elementary particles is related to infinitesimal rotations. [2]
The splitting of the inner product of a complex Hilbert space into real part and complex part corresponds to a splitting of the unitary geometry in Hilbert spaces into symplectic geometry and K¨ahler geometry. These geometries are fundamental for both classical and modern physics. [2]
The starting point for linear and multilinear algebra is the problem of solving linear systems of equations. [3]
The statistical physics of the multi-particle system of N harmonic oscillators is governed by the Euclidean propagator of a single harmonic oscillator. [2]
The symmetry behind renormalization theory can be described by Hopf algebras. [2]
The theory of determinants is equivalent to the theory of volume functions. [3]
The theory of distributions is based on duality. Costates are dual states. [2]
The trace of linear operators is critically used in statistical physics. [3]
The trick is to replace the free photon quantum field by the wave functions of bound electrons. [2]
The use of the Moyal product for smooth functions avoids the use of Hilbert-space operators in quantum mechanics. [2]
Thermodynamical systems in equilibrium states (e.g., chemical substances) can be described by the Gibbs contact form and its integral manifolds. [2]
They are obtained by deforming the coordinate Hopf algebra of a group. [3]
Topology has its roots in geometric intuition, the theory of analytic functions, the theory of Abelian integrals over algebraic functions, and in physics. [2]
Topology is rooted in Maxwell’s theory on the electromagnetic field. [1]
Try to dualize in mathematics as much as you can. [2]
Two continuous mappings are contained in the same mapping class iff they can be continuously deformed into each other. In important special cases, the space of mapping classes can be equipped with an additional group structure. This leads to Poincaré’s fundamental group and the higher homotopy groups of topological spaces. [2]
Two-dimensional spheres are the simplest curved surfaces. They serve as prototypes for the geometry and analysis of manifolds. [2]
Unfortunately, the computations of radiative corrections in quantum electrodynamics are rather lengthy. For getting high accuracy by including multi-loop corrections, one needs sophisticated supercomputer programs. [2]
Use tensor products and Fock spaces for describing mathematically the states of many-particle systems. [2]
Wave operators describe the motion of wave packets; the scattering operator is closely related to the S-matrix. [2]
Waves are used in nature in order to transport energy and information. [2]
Whoever understands creation and annihilation operators can understand everything in quantum physics. [2]
Whoever understands Feynman diagrams can understand everything in quantum field theory. [2]
Whoever understands Green’s functions can understand forces in nature. [1]
Whoever understands symmetries can understand everything in this world. [2]
Whoever understands the quantization of the harmonic oscillator can understand everything in quantum physics. [2]
Whoever understands the S-matrix (scattering matrix) can understand everything in the theory of scattering processes for elementary particles. [2]
Sources:
[1] Eberhard Zeidler (2006) "Quantum Field Theory I: Basics in Mathematics and Physics"
[2] Eberhard Zeidler (2006) "Quantum Field Theory II: Quantum Electrodynamics"
[3] Eberhard Zeidler (2006) "Quantum Field Theory I: Gauge Theory"
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