Browse All Assumptions

About Assumptions

Every physics derivation rests on a foundation of assumptions. The TheorIA Dataset maintains a centralized database of assumptions that are referenced across all entries, ensuring consistent terminology and clear dependencies.

Assumptions are categorized into three types:

Each assumption can be referenced by multiple entries, and clicking on the "Used in" links shows which physics results depend on that particular assumption.

Principles

Canonical Commutation Relations

Canonical variables `q_i` and `p_i`, representing generalized coordinates and their conjugate momenta for each degree of freedom, satisfy the commutation relations `[q_i, p_j] = i*ℏ*delta_(ij) `, `[q_i, q_j] = 0`, and `[p_i, p_j] = 0`. These relations encode the basic quantum kinematics of canonical pairs and fix the fundamental scale through `ℏ`.

Mathematical Form:

`[q_i, p_j] = i*ℏ*delta_(ij)`
`[q_i, q_j] = 0`
`[p_i, p_j] = 0`

Symbols:

`q_i`: Generalized coordinate operator for the `i`-th degree of freedom.
`p_i`: Momentum operator canonically conjugate to `q_i`.
`ℏ`: Reduced Planck constant, `ℏ = h/(2*pi)`.
`delta_(ij)`: Kronecker delta (`delta_(ij) = 1` if `i = j`, `0` otherwise).
Used in: Uncertainty Principle, Ladder Operators
PRINCIPLE

Canonical Field Anticommutation Relations

For a Dirac spinor field `psi_alpha(vec x, t)` and its conjugate momentum density `pi_alpha(vec x, t) = i*psi_alpha^(dagger)(vec x, t)`, the equal-time anticommutation relations are `{psi_alpha(vec x, t), psi_beta^(dagger)(vec y, t)} = delta_(alpha beta)*delta^3(vec x - vec y)`, `{psi_alpha(vec x, t), psi_beta(vec y, t)} = 0`, and `{psi_alpha^(dagger)(vec x, t), psi_beta^(dagger)(vec y, t)} = 0`. This is the fermionic analog of canonical commutation relations, required by the spin-statistics theorem for half-integer spin fields.

Mathematical Form:

`{psi_alpha(vec x, t), psi_beta^(dagger)(vec y, t)} = delta_(alpha beta)*delta^3(vec x - vec y)`
`{psi_alpha(vec x, t), psi_beta(vec y, t)} = 0`
`{psi_alpha^(dagger)(vec x, t), psi_beta^(dagger)(vec y, t)} = 0`

Symbols:

`psi_alpha(vec x, t)`: Dirac spinor field operator component `alpha` at position `vec x` and time `t`.
`psi_alpha^(dagger)(vec x, t)`: Hermitian conjugate of the Dirac spinor field component.
`{A, B}`: Anticommutator of operators, `{A, B} = A*B + B*A`.
`delta_(alpha beta)`: Kronecker delta for spinor indices.
`delta^3(vec x - vec y)`: Three-dimensional Dirac delta function.
Used in: Dirac Field Quantization
PRINCIPLE

Canonical Field Commutation Relations

For a scalar field `phi(vec x, t)` and its conjugate momentum density `pi(vec x, t)`, the equal-time commutation relations are `[phi(vec x, t), pi(vec y, t)] = i*ℏ*delta^3(vec x - vec y)`, `[phi(vec x, t), phi(vec y, t)] = 0`, and `[pi(vec x, t), pi(vec y, t)] = 0`. This is the field-theoretic generalization of canonical commutation relations, where the Kronecker delta becomes a Dirac delta function in the continuum limit.

Mathematical Form:

`[phi(vec x, t), pi(vec y, t)] = i*ℏ*delta^3(vec x - vec y)`
`[phi(vec x, t), phi(vec y, t)] = 0`
`[pi(vec x, t), pi(vec y, t)] = 0`

Symbols:

`phi(vec x, t)`: Scalar field operator at position `vec x` and time `t`.
`pi(vec x, t)`: Conjugate momentum density operator.
`delta^3(vec x - vec y)`: Three-dimensional Dirac delta function.
`ℏ`: Reduced Planck constant.
Used in: Scalar Field Quantization
PRINCIPLE

Classical Electromagnetism Framework

Fields and sources are classical, smooth (continuously differentiable, `C^1` or better), and defined on flat Minkowski spacetime. Vector calculus theorems (divergence theorem, Stokes' theorem) apply. Quantum and gravitational effects are negligible.

Used in: Maxwell Equations
PRINCIPLE

Conservation Laws

Total energy, linear momentum and angular momentum are conserved.

Mathematical Form:

`(dE)/(dt) = 0`
`(dvec{p})/(dt) = 0`
`(dvec{L})/(dt) = 0`

Symbols:

`E`: Total energy of the system
`vec{p}`: Total momentum vector of the system
`vec{L}`: Total angular momentum vector of the system
`t`: Time
Used in: Vis Viva, Relativistic Energy Momentum
PRINCIPLE

Correspondence Principle

Quantum mechanics must reproduce classical mechanics in the appropriate limit (large quantum numbers, macroscopic scales, or formally `ℏ -> 0`). In particular, the quantum Hamiltonian operator is chosen so that its spectrum and dynamics reduce to those of the classical Hamiltonian `H_("classical")(p, r, t)` when quantum effects become negligible.

Mathematical Form:

`lim_{ℏ -> 0} ("quantum predictions") = ("classical predictions")`
`H_("classical")(p, r, t) = p^2/(2*m) + V(r, t)`

Symbols:

`ℏ`: Reduced Planck constant
`H_("classical")`: Classical Hamiltonian function built from position, momentum and potential energy
`p`: Classical particle momentum
`r`: Classical position
`V(r, t)`: Classical scalar potential energy as a function of position and time
Used in: Schrodinger Equation
PRINCIPLE

Electromagnetic Field Quantization

The electromagnetic field is quantized, with each mode having discrete energy levels. Each mode of frequency `nu` can only have energies that are integer multiples of `h*nu`, where `h` is Planck's constant and the integer represents the photon occupation number.

Mathematical Form:

`E_n = n*h*nu`

Symbols:

`E_n`: Energy of mode with occupation number n
`n`: Photon occupation number (non-negative integer)
`h`: Planck's constant
`nu`: Frequency of the electromagnetic mode
Used in: Blackbody Radiation
PRINCIPLE

Existence of a Poincaré-Invariant Vacuum

There exists a unique (up to phase) vacuum state `|:0:)` invariant under spacetime symmetries, from which particle states are obtained by acting with local fields.

Mathematical Form:

`U(g)*|:0:) = |:0:), forall g`

Symbols:

`|:0:)`: Vacuum state.
`U(g)`: Unitary representation of a spacetime symmetry `g`.
Used in: Spin Statistics Theorem, Fock Space, Scalar Field Quantization
PRINCIPLE

Gleason Theorem Conditions

The Hilbert space has dimension at least 3 (`dim(H) >= 3`), and the probability assignment on projectors (orthogonal projection operators `P` satisfying `P^2 = P = P^ast`) is continuous

Mathematical Form:

`dim(H) >= 3`

Symbols:

`H`: Complex Hilbert space of quantum states
`dim(H)`: Dimension of the Hilbert space
`P`: Projector (orthogonal projection operator) on `H`, satisfying idempotence and self-adjointness
Used in: Born Rule
PRINCIPLE

Hilbert Space Probability Structure

Quantum measurement outcomes correspond to orthogonal projection operators (projectors) on a complex Hilbert space `H`, where projectors are self-adjoint operators satisfying `P^2 = P = P^ast`. Probabilities can be assigned to these projectors via a probability measure `mu` mapping projectors to real numbers in `[0,1]`.

Mathematical Form:

`P^2 = P = P^ast`
`mu: {P} -> [0,1]`

Symbols:

`H`: Complex Hilbert space of quantum states
`P`: Projector (orthogonal projection operator) on `H`, satisfying idempotence and self-adjointness
`mu`: Probability measure mapping projectors to real numbers in `[0,1]`
Used in: Born Rule, Schrodinger Equation, Uncertainty Principle, Dirac Equation, Ladder Operators
PRINCIPLE

Inertial Reference Frame

Observations are made from an inertial (non-accelerating) reference frame

Used in: Special Relativity Transformations, Lorentz Group And Four Vectors
PRINCIPLE

Lorentz Covariance

Spacetime symmetries act consistently on both coordinates and quantum states. For every proper orthochronous Lorentz transformation `Lambda` (a linear map of spacetime that preserves the Minkowski interval), there exists a unitary operator `U(Lambda)` on the Hilbert space such that local fields transform covariantly: the field at point `x` in one frame is related to the field at the transformed point `Lambda*x` in the new frame, with its field indices rotated by the appropriate finite-dimensional Lorentz representation `D(Lambda)` (scalar/vector/spinor, etc.).

Mathematical Form:

`Lambda^T*eta*Lambda = eta`
`det(Lambda) = 1`
`Lambda_0^0 >= 1`
`U(Lambda)*Phi_a(x)*U(Lambda)^(-1) = sum_b D(Lambda)_a^b*Phi_b(Lambda*x)`

Symbols:

`x`: Spacetime point (4-vector), e.g. `x = (c*t, x, y, z)`.
`eta`: Minkowski metric tensor.
`Lambda`: Proper orthochronous Lorentz transformation (a real `4x4` matrix) preserving the Minkowski interval: `Lambda^T*eta*Lambda = eta`, with `det(Lambda)=1` and `Lambda_0^0 >= 1`.
`U(Lambda)`: Unitary operator implementing the Lorentz transformation `Lambda` on the Hilbert space of states.
`Phi_a(x)`: Local field operator evaluated at spacetime point `x`, with component/field index `a`.
`D(Lambda)_a^b`: Finite-dimensional Lorentz representation acting on the field indices (e.g. `D=1` for a scalar, `D=Lambda` for a 4-vector, and `D=S(Lambda)` for a spinor).
Used in: Spin Statistics Theorem, Klein Gordon Lagrangian
PRINCIPLE

Microcausality (Locality)

Microcausality is the quantum-field-theoretic formulation of relativistic causality. It states that local observables associated with spacelike-separated spacetime points correspond to mutually compatible measurements. If two spacetime points cannot be connected by signals propagating at or below the speed of light, then operations performed at one point cannot influence physical outcomes at the other. This requirement forbids superluminal signaling and ensures causal consistency of relativistic quantum theories.

Mathematical Form:

`(x - y)^2 < 0 => [O(x), O(y)] = 0`

Symbols:

`O(x)`: Local observable or field operator localized at the spacetime point `x`, constructed from quantum fields and acting nontrivially only in an arbitrarily small neighborhood of `x`.
`[A,B]`: Commutator of two operators, defined as `[A,B] = A*B - B*A`. Vanishing commutator implies that the corresponding measurements are compatible and order-independent.
`(x - y)^2`: Minkowski squared spacetime interval between points `x` and `y`, defined in the `(+,-,-,-)` metric convention as `(x - y)^2 = c^2*(t_x - t_y)^2 - |x_vec - y_vec|^2`. A negative value indicates spacelike separation.
`c`: Speed of light in vacuum, setting the maximum speed for causal signal propagation.
Used in: Spin Statistics Theorem
PRINCIPLE

Planck-de Broglie Relations

Energy and momentum of matter waves are related to frequency and wavevector by `E = ℏ*omega` and `p = ℏ*k`.

Mathematical Form:

`E = ℏ*omega`
`p = ℏ*k`

Symbols:

`E`: Energy of the quantum state
`p`: Magnitude of particle momentum
`ℏ`: Reduced Planck constant
`omega`: Angular frequency of the matter wave
`k`: Magnitude of wavevector of the matter wave
Used in: Schrodinger Equation, Klein Gordon Equation
PRINCIPLE

Positive Energy Spectrum (Spectrum Condition)

The energy-momentum spectrum is bounded below and lies in the forward light cone: physical states have `p^0 >= 0`.

Mathematical Form:

`p^0 >= 0`

Symbols:

`p^0`: Energy component of the four-momentum.
Used in: Spin Statistics Theorem
PRINCIPLE

Quantum Observables as Self-Adjoint Operators

Each physical observable is represented by a self-adjoint (Hermitian) operator `A` on the Hilbert space, so expectation values like `(:psi:|A|:psi:)` are real and variances defined as `Delta A^2 = (:psi:|(A - a_0)^2|:psi:)` with `a_0 = (:psi:|A|:psi:)` are non-negative.

Mathematical Form:

`a_0 = (:psi:|A|:psi:)`
`Delta A^2 = (:psi:|(A - a_0)^2|: psi:)`

Symbols:

`A`: Self-adjoint operator representing a physical observable.
`|:psi:)`: Normalized state vector in the quantum Hilbert space.
`a_0`: Expectation value of observable `A` in state `|:psi:)`, `a_0 = (:psi:|A|:psi:)`.
`Delta A^2`: Variance (squared uncertainty) of `A` in state `|:psi:)`.
Used in: Uncertainty Principle
PRINCIPLE

Quantum Superposition and Linear Dynamics

Quantum states form a complex vector space and physically allowed time evolutions act linearly on this space. If `psi_1` and `psi_2` are solutions of the evolution equation, any complex linear combination `alpha*psi_1 + beta*psi_2` is also a solution.

Mathematical Form:

`psi(t) in H`
`psi(t) = alpha*psi_1(t) + beta*psi_2(t) => psi(t) " solves the same evolution equation"`

Symbols:

`H`: Complex Hilbert space of quantum states
`psi`: Quantum state (e.g. wavefunction in a chosen representation)
`alpha, beta`: Complex coefficients describing superpositions
Used in: Schrodinger Equation
PRINCIPLE

Stationary Action Principle

The system obeys the principle of stationary action (least action principle), meaning the path taken by the system makes the action stationary, leading to the Euler-Lagrange equations

Used in: Noethers Theorem, Klein Gordon Lagrangian
PRINCIPLE

Variational Calculus Framework

System describable by generalized coordinates `q_i(t)` with well-defined, twice-differentiable Lagrangian `L(q_i, dot q_i, t)`, smooth trajectories, and suitable boundary conditions for variational analysis (fixed endpoints, allowable interior variations)

Mathematical Form:

`q_i = q_i(t)`
`L = L(q_i, dot q_i, t)`
`delta q_i(t_1) = delta q_i(t_2) = 0`

Symbols:

`q_i`: Generalized coordinates describing the system configuration
`dot q_i`: Time derivatives of generalized coordinates (generalized velocities)
`L`: Lagrangian function of the system
`t`: Time parameter
`delta q_i`: Variation of generalized coordinates
`t_1, t_2`: Initial and final times (fixed endpoints)
Used in: Least Action Principle
PRINCIPLE

Empirical

Ampère-Biot-Savart Law

Steady electric currents generate magnetic fields that circulate around them. The line integral of magnetic field around a closed loop equals `mu_0` times the enclosed current.

Mathematical Form:

`oint_C B cdot dl = mu_0 * I_(enc)`

Symbols:

`B`: Magnetic field
`mu_0`: Magnetic permeability of free space
`I_(enc)`: Current enclosed by the loop
Used in: Maxwell Equations
EMPIRICAL

Coulomb's Law

Electric charges interact via an inverse-square force law. A point charge `q` produces an electric field `E(r) = q / (4 * pi * epsilon_0 * r^2) * hat(r)` where `hat(r)` is the radial unit vector.

Mathematical Form:

`E(r) = q / (4 * pi * epsilon_0 * r^2) * hat(r)`

Symbols:

`E`: Electric field vector
`q`: Electric point charge
`epsilon_0`: Electric permittivity of free space
`r`: Distance from the point charge
`hat(r)`: Radial unit vector
Used in: Maxwell Equations
EMPIRICAL

Electromagnetic Polarization

Electromagnetic waves have two independent polarization directions (transverse to propagation direction)

Used in: Blackbody Radiation
EMPIRICAL

Faraday's Law of Electromagnetic Induction

A time-varying magnetic flux through a closed circuit induces an electromotive force (emf) in that circuit. The induced emf equals the negative rate of change of magnetic flux.

Mathematical Form:

`text{emf} = -(d Phi_B)/(dt)`

Symbols:

`text{emf}`: Electromotive force
`Phi_B`: Magnetic flux through a surface
Used in: Maxwell Equations
EMPIRICAL

Light Speed Constant

The speed of light in vacuum is constant and independent of the motion of the source or observer

Mathematical Form:

`c = 299792458 text{ m/s}`

Symbols:

`c`: Speed of light in vacuum, exact value by definition in SI units.
Used in: Relativistic Energy Momentum, Special Relativity Transformations, Lorentz Group And Four Vectors
EMPIRICAL

No Magnetic Monopoles

Magnetic monopoles (isolated magnetic charges) have never been observed experimentally. As a consequence, the net magnetic flux through any closed surface is zero. (no free magnetic charge density exists).

Mathematical Form:

`oint_S B * dA = 0`

Symbols:

`B`: Magnetic flux density
Used in: Maxwell Equations
EMPIRICAL

Well-Defined Rest Mass

Particles have well-defined rest masses that do not depend on the reference frame

Used in: Relativistic Energy Momentum
EMPIRICAL

Approximations

Classical Macroscopic Limit

For macroscopic bodies with de Broglie wavelength `λ_(dB) < < L`, quantum effects such as interference and wave-particle duality are negligible and the motion can be described by classical trajectories.

Mathematical Form:

`lambda_(dB) < < L`

Symbols:

`lambda_(dB)`: De Broglie wavelength of the body
`L`: Characteristic length scale of the system or apparatus
Used in: Special Relativity Transformations
APPROXIMATION

Constant Relative Velocity

Reference frames move with constant relative velocity (no acceleration)

Mathematical Form:

`(dv)/(dt) = 0`

Symbols:

`v`: Relative velocity between reference frames
`t`: Time
Used in: Special Relativity Transformations
APPROXIMATION

Flat Spacetime

Spacetime is flat, no gravitational effects considered

Mathematical Form:

`g_(mu nu) = eta_(mu nu) = diag(+1,-1,-1,-1)`

Symbols:

`g_(mu nu)`: Spacetime metric tensor in general.
`eta_(mu nu)`: Minkowski metric components used in special relativity, with signature `(+,-,-,-)`.
`mu, nu`: Lorentz indices running over `0,1,2,3` (Einstein summation on repeated indices).
Used in: Relativistic Energy Momentum, Lorentz Group And Four Vectors, Klein Gordon Equation, Dirac Equation, Klein Gordon Lagrangian
APPROXIMATION

Linear Isotropic Vacuum

The medium is vacuum (free space) characterized by constants `epsilon_0` (electric permittivity) and `mu_0` (magnetic permeability), but may contain charge density `rho(r,t)` and current density `J(r,t)`. The vacuum is linear, homogeneous, and isotropic with constitutive relations `D = epsilon_0 * E` and `H = B / mu_0`.

Mathematical Form:

`D = epsilon_0 * E`
`H = B / mu_0`

Symbols:

`epsilon_0`: Electric permittivity of free space
`mu_0`: Magnetic permeability of free space
`D`: Electric displacement field
`E`: Electric field
`H`: Magnetic field intensity
`B`: Magnetic flux density
Used in: Maxwell Equations
APPROXIMATION

Non-interacting particle approximation

Particles are assumed not to interact. The total energy of a many-particle configuration is additive in the single-particle level occupations, and in the grand-canonical ensemble the partition function factorizes into independent contributions from each single-particle level.

Mathematical Form:

`E({n_r}) = sum_r n_r*epsilon_r`
`Z = prod_r Z_r`

Symbols:

`r`: Index labeling single-particle energy levels (modes).
`n_r`: Number of particles occupying the single-particle level labeled by `r`.
`E({n_r})`: Total energy of a many-particle configuration specified by occupation numbers `{n_r}`.
`epsilon_r`: Energy eigenvalue of the single-particle level labeled by `r`.
`Z`: Grand partition function of the full system.
`Z_r`: Single-level contribution to the grand partition function associated with level `r`.
Used in: Spin Statistics Theorem
APPROXIMATION

Nonrelativistic Regime

Particle velocities satisfy `v < < c` so that relativistic corrections to the kinetic energy and dynamics can be neglected. In this limit the kinetic energy is well approximated by `E ≃ p^2/(2*m)`.

Mathematical Form:

`v < < c`
`E ≃ p^2/(2*m)`

Symbols:

`v`: Characteristic particle velocity
`c`: Speed of light in vacuum
`E`: Particle energy
`p`: Magnitude of particle momentum
`m`: Particle mass
Used in: Keplers Laws, Schrodinger Equation
APPROXIMATION

Perfect Blackbody

The spectral energy density `u(nu,T)` describes radiation from a perfect black body - an idealized emitter that absorbs all incident radiation and re-emits it purely based on temperature `T`

Used in: Blackbody Radiation
APPROXIMATION

Point Mass Approximation

Both bodies can be treated as point masses with spherically symmetric mass distributions

Used in: Keplers Laws, Vis Viva
APPROXIMATION

System Isolation

System is sufficiently isolated from external influences (for gravitational systems: negligible external perturbations)

Used in: Keplers Laws, Vis Viva
APPROXIMATION

Thermal Equilibrium

The radiation field is in thermal equilibrium at temperature `T`, meaning the emission and absorption rates are balanced and the energy distribution is time-independent

Used in: Blackbody Radiation, Partition Function
APPROXIMATION