Crate deep_causality_physics Copy item path Source beam diffraction grmhd ideal moire phase physics_error plasma polarization qgt ray resistive AbcdMatrix ABCD Matrix. $2 \times 2$ Ray Transfer Matrix. Acceleration Linear acceleration (m/s^2).
Can be negative to indicate direction in 1D context. Activity Radioactivity (Becquerels). AlfvenSpeed Alfven Speed ($v_A$). Characteristic speed of magnetic waves in plasma.
Unit: m/s. Constraint: >= 0. AmountOfSubstance Amount of Substance (Moles). Area Area (m^2). BandDrudeWeight Band Drude Weight ($D$). BeamWaist Beam Waist ($w_0$). Minimum radius of Gaussian beam.
Unit: Meters. Constraint: > 0. BerryCurvature Berry Curvature component ($Ω_{ij}$). ChemicalPotentialGradient Chemical Potential Gradient $\nabla μ$. ComplexBeamParameter Complex Beam Parameter ($q(z) = z + i z_R$).
Constraint: $\text{Im}(q) > 0$. Concentration Concentration field $c(\mathbf{r})$. Conductance Electrical Conductance ($G$). Conductivity Electrical Conductivity ($\sigma$).
Unit: Siemens/m (S/m). Constraint: > 0. DebyeLength Debye Length ($\lambda_D$). Screening length in plasma.
Unit: Meters (m). Constraint: > 0. Density Density (kg/m^3). Diffusivity Magnetic Diffusivity ($\eta$).
Unit: $m^2/s$. Constraint: >= 0. Displacement Displacement field $\mathbf{u}(\mathbf{r})$. Efficiency Thermodynamic efficiency (0.0 to 1.0). ElectricPotential Electric Potential (Volts or J/C). Energy Energy (Joules or eV). EnergyDensity Energy Density (Joules per cubic meter). Entropy Entropy (J/K). FocalLength Focal Length ($f$).
Unit: Meters. Constraint: None (can be negative for diverging lens). Force Force (N).
Can be negative to indicate direction in 1D context. Frequency Frequency (Hz). HalfLife Half-Life (Seconds). IndexOfRefraction Index of refraction for a medium (ratio of c to phase velocity). JonesVector Jones Vector. Polarized Electric Field. Rank 1, Dim 2 Complex Tensor. LarmorRadius Larmor Radius ($r_L$). Gyroradius of a charged particle.
Unit: Meters (m). Constraint: > 0. Length Length (m). MagneticFlux Magnetic Flux (Webers). MagneticPressure Magnetic Pressure ($P_B$). Energy density of the magnetic field.
Unit: Pascals (Pa). Constraint: >= 0. Mass Mass quantity (kg). MaxwellSolver A standard solver for Maxwell’s Equations in Geometric Algebra. Mobility Charge Carrier Mobility ($μ$). MomentOfInertia Moment of Inertia (kg·m^2). Momentum Momentum vector $\mathbf{k}$. NumericalAperture Numerical Aperture ($NA = n \sin \theta$).
Unit: Dimensionless. Constraint: > 0. OpticalPower Optical Power ($D = 1/f$).
Unit: Diopters ($m^{-1}$). Constraint: None. OrbitalAngularMomentum Orbital Angular Momentum ($L$). OrderParameter Superconducting Order Parameter ($ψ$). PhaseAngle Phase Angle (Radians). PhysicalField Wrapper for CausalMultiVector representing a physical field (E, B, etc.).
Implements Default to return a zero vector. PhysicalVector PhysicsError PlasmaBeta Plasma Beta ($\beta$). Ratio of thermal to magnetic pressure.
Unit: Dimensionless. Constraint: >= 0. PlasmaFrequency Plasma Frequency ($\omega_{pe}$). Natural oscillation frequency.
Unit: Rad/s. Constraint: > 0. Pressure Pressure (Pascals). Probability Probability value [0, 1]. QuantumEigenvector Quantum Eigenvector $|u_n➢$. QuantumMetric Quantum Metric component ($g_{ij}$). QuantumVelocity Quantum Velocity vector $\partial_i H |u_n➢$. Ratio A generic dimensionless ratio. RayAngle Ray Angle ($\theta$). Angle relative to optical axis.
Unit: Radians. RayHeight Ray Height ($y$). Distance from optical axis.
Unit: Meters. SpacetimeInterval Spacetime Interval ($s^2$). SpacetimeVector Wrapper for CausalMultiVector representing a vector in Spacetime. Speed Speed quantity (scalar magnitude of velocity) (m/s). Stiffness Scalar stiffness (Young’s Modulus, etc.) (Pascals). StokesVector Stokes Vector. Intensity vector $(S_0, S_1, S_2, S_3)$. Rank 1, Dim 4 Tensor.
Constraint: $S_0^2 \ge S_1^2 + S_2^2 + S_3^2$. Stress Scalar stress (Pascals), used for simple 1D cases or invariants (Von Mises). Temperature Absolute Temperature (Kelvin). Time Time (Seconds). Torque Torque (N·m).
Can be negative to indicate direction (e.g. clockwise vs counter-clockwise). TwistAngle Moiré Twist Angle ($θ$). VectorPotential Electromagnetic Vector Potential $\mathbf{A}$. Viscosity Dynamic Viscosity (Pa·s). Volume Volume (m^3). Wavelength Wavelength ($\lambda$).
Unit: Meters. Constraint: > 0. PhysicsErrorEnum Detailed classification of physics-related errors. ATOMIC_MASS_CONSTANT AVOGADRO_CONSTANT BOLTZMANN_CONSTANT ELECTRON_MASS ELEMENTARY_CHARGE FINE_STRUCTURE_CONSTANT G GRAPHENE_LATTICE_CONST NEWTONIAN_CONSTANT_OF_GRAVITATION PLANCK_CONSTANT PROTON_MASS REDUCED_PLANCK_CONSTANT SPEED_OF_LIGHT STEFAN_BOLTZMANN_CONSTANT VACUUM_ELECTRIC_PERMITTIVITY VACUUM_MAGNETIC_PERMEABILITY QuantumGates Standard Quantum Gates interface. QuantumOps Core Quantum State Operations (Dirac Notation). alfven_speed alfven_speed_kernel Calculates the characteristic speed of Alfven waves.
$$ v_A = \frac{B}{\sqrt{\mu_0 \rho}} $$ angular_momentum Causal wrapper for kinematics::angular_momentum_kernel angular_momentum_kernel Calculates angular momentum as the outer product of radius and linear momentum: $L = r \wedge p$. apply_gate Causal wrapper for mechanics::apply_gate_kernel apply_gate_kernel Applies a quantum gate to a state: $|\psi’\rangle = U |\psi\rangle$. beam_spot_size beam_spot_size_kernel Extracts the beam spot size $w(z)$ from the complex beam parameter $q$. bernoulli_pressure Causal wrapper for mechanics::bernoulli_pressure_kernel bernoulli_pressure_kernel Calculates pressure $P_2$ using Bernoulli’s principle. binding_energy Causal wrapper for physics::binding_energy_kernel binding_energy_kernel Calculates nuclear binding energy (mass defect): $E = \Delta m c^2$. bistritzer_macdonald Wrapper for moire::bistritzer_macdonald_kernel bistritzer_macdonald_kernel Constructs the Bistritzer-MacDonald Continuum Hamiltonian for Twisted Bilayer Graphene (TBG). boltzmann_factor Causal wrapper for stats::boltzmann_factor_kernel boltzmann_factor_kernel Calculates the unnormalized Boltzmann factor: $e^{-E/k_BT}$. born_probability Causal wrapper for mechanics::born_probability_kernel born_probability_kernel Calculates the Born probability: $P = |\langle \text{basis} | \text{state} \rangle|^2$. cahn_hilliard_flux Wrapper for phase::cahn_hilliard_flux_kernel cahn_hilliard_flux_kernel Calculates the Cahn-Hilliard Flux with degenerate mobility (Type B). carnot_efficiency Causal wrapper for stats::carnot_efficiency_kernel carnot_efficiency_kernel Calculates Carnot Efficiency: $\eta = 1 - \frac{T_C}{T_H}$. chronometric_volume Causal wrapper for spacetime::chronometric_volume_kernel chronometric_volume_kernel Calculates Chronometric Volume (4-Volume) from 3 vectors? commutator Causal wrapper for mechanics::commutator_kernel commutator_kernel Calculates commutator $[A, B] = AB - BA$. debye_length debye_length_kernel Calculates the Debye Length $\lambda_D$.
$$ \lambda_D = \sqrt{\frac{\epsilon_0 k_B T_e}{n_e e^2}} $$ degree_of_polarization degree_of_polarization_kernel Calculates the Degree of Polarization (DOP) from a Stokes vector. doppler_effect_approaching Monadic wrapper for general::doppler_effect_kernel doppler_effect_kernel Calculates the observed frequency due to the Doppler effect for longitudinal motion. effective_band_drude_weight Wrapper for qgt::effective_band_drude_weight_kernel effective_band_drude_weight_kernel Approximates the Effective Band Drude Weight ($D$) using band curvature and quantum geometry. einstein_tensor Causal wrapper for gravity::einstein_tensor_kernel einstein_tensor_kernel Calculates the Einstein Tensor: $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}$. energy_momentum_tensor_em energy_momentum_tensor_em_kernel Calculates the electromagnetic stress-energy tensor $T^{\mu\nu}{EM}$.
$$ T^{\mu\nu} = F^{\mu\alpha}F^\nu \alpha - \frac{1}{4} g^{\mu\nu} F_{\alpha\beta}F^{\alpha\beta} $$ escape_velocity Causal wrapper for mechanics::escape_velocity_kernel escape_velocity_kernel Calculates escape velocity: $v_e = \sqrt{\frac{2GM}{r}}$. expectation_value Causal wrapper for mechanics::expectation_value_kernel expectation_value_kernel Calculates the expectation value: $\langle A \rangle = \langle \psi | A | \psi \rangle$. fidelity Causal wrapper for mechanics::fidelity_kernel fidelity_kernel Calculates Quantum Fidelity: $F = |\langle \psi_{\text{ideal}} | \psi_{\text{actual}} \rangle|^2$. foppl_von_karman_strain Wrapper for moire::foppl_von_karman_strain_kernel foppl_von_karman_strain_kernel Calculates the Föppl-von Kármán stress tensor using the complete theory (Field-based). foppl_von_karman_strain_simple Wrapper for moire::foppl_von_karman_strain_simple_kernel foppl_von_karman_strain_simple_kernel Calculates the Föppl-von Kármán stress tensor (Simplified / Local). gaussian_q_propagation gaussian_q_propagation_kernel Propagates a Gaussian beam’s complex $q$-parameter through an ABCD optical system. generalized_master_equation Causal wrapper for estimation::generalized_master_equation_kernel generalized_master_equation_kernel Generalized Master Equation Kernel. generate_schwarzschild_metric Generates a Schwarzschild-like 4D metric tensor. geodesic_deviation Causal wrapper for gravity::geodesic_deviation_kernel geodesic_deviation_kernel Calculates Geodesic Deviation acceleration: $A^\mu = -R^\mu_{\nu\sigma\rho} V^\nu n^\sigma V^\rho$. ginzburg_landau_free_energy Wrapper for phase::ginzburg_landau_free_energy_kernel ginzburg_landau_free_energy_kernel Calculates the Ginzburg-Landau Free Energy density. grating_equation grating_equation_kernel Calculates the diffraction angle for a Grating using the Grating Equation. hadronization Causal wrapper for physics::hadronization_kernel hadronization_kernel Simulates phenomenological hadronization: $E_{\text{density}} \rightarrow \vec{p}_{\text{jet}}$. haruna_cz_gate Causal wrapper for mechanics::haruna_cz_gate_kernel haruna_cz_gate_kernel Implements Haruna’s Logical CZ Gate. haruna_hadamard_gate Causal wrapper for mechanics::haruna_hadamard_gate_kernel haruna_hadamard_gate_kernel Implements Haruna’s Logical Hadamard Gate. haruna_s_gate Causal wrapper for mechanics::haruna_s_gate_kernel haruna_s_gate_kernel Implements Haruna’s Logical S-Gate. haruna_t_gate Causal wrapper for mechanics::haruna_t_gate_kernel haruna_t_gate_kernel Implements Haruna’s Logical T-Gate. haruna_x_gate Causal wrapper for mechanics::haruna_x_gate_kernel haruna_x_gate_kernel Implements Haruna’s Logical X-Gate. haruna_z_gate Causal wrapper for mechanics::haruna_z_gate_kernel haruna_z_gate_kernel Implements Haruna’s Logical Z-Gate. heat_capacity Causal wrapper for stats::heat_capacity_kernel heat_capacity_kernel Calculates Heat Capacity: $C = \frac{dE}{dT}$. heat_diffusion Causal wrapper for stats::heat_diffusion_kernel heat_diffusion_kernel Calculates the Heat Equation step: $\frac{\partial u}{\partial t} = \alpha \Delta u$. hookes_law Causal wrapper for mechanics::hookes_law_kernel hookes_law_kernel Calculates generalized Hooke’s Law: $\sigma_{ij} = C_{ijkl} \epsilon_{kl}$. hydrostatic_pressure Causal wrapper for mechanics::hydrostatic_pressure_kernel hydrostatic_pressure_kernel Calculates hydrostatic pressure: $P = P_0 + \rho g h$. ideal_gas_law Causal wrapper for stats::ideal_gas_law_kernelRatio (or scalar). ideal_gas_law_kernel Calculates the Ideal Gas Constant $R$ from state variables: $R = \frac{PV}{nT}$. ideal_induction ideal_induction_kernel Calculates the time evolution of the magnetic field (Frozen-in flux).
$$ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) $$ jones_rotation jones_rotation_kernel Rotates a Jones Matrix (operator) by an angle $\phi$. kalman_filter_linear Causal wrapper for estimation::kalman_filter_linear_kernel kalman_filter_linear_kernel Standard Linear Kalman Filter Update Step. kinetic_energy Causal wrapper for kinematics::kinetic_energy_kernel kinetic_energy_kernel Calculates translational kinetic energy: $K = \frac{1}{2} m v^2$. klein_gordon Causal wrapper for mechanics::klein_gordon_kernel klein_gordon_kernel Calculates the Klein-Gordon operator action: $(\Delta + m^2)\psi$. larmor_radius larmor_radius_kernel Calculates the Larmor Radius (Gyroradius).
$$ r_L = \frac{m v_\perp}{|q| B} $$ lens_maker lens_maker_kernel Calculates optical power and focal length using the Lens Maker’s Equation. logical_cz Calculates the Logical CZ gate: $CZ(\gamma_1, \gamma_2) = \exp(i \pi a(\gamma_1) a(\gamma_2))$. logical_hadamard Calculates the Logical Hadamard gate. logical_s Calculates the Logical S gate: $S(\gamma) = \exp(i \frac{\pi}{2} a(\gamma)^2)$. logical_t Calculates the Logical T gate. logical_x Calculates the Logical X gate: $X(\tilde{\gamma}) = \exp(i \pi b(\tilde{\gamma}))$. logical_z Calculates the Logical Z gate: $Z(\gamma) = \exp(i \pi a(\gamma))$. lorentz_force Causal wrapper for forces::lorentz_force_kernel lorentz_force_kernel Calculates the magnetic force component of the Lorentz Force: $F = J \times B$. lorenz_gauge Causal wrapper for fields::lorenz_gauge_kernel lorenz_gauge_kernel Calculates the Lorenz gauge condition: $\nabla \cdot A = 0$. magnetic_helicity_density Causal wrapper for fields::magnetic_helicity_density_kernel magnetic_helicity_density_kernel Calculates Magnetic Helicity Density: $h = A \cdot B$. magnetic_pressure magnetic_pressure_kernel Calculates magnetic pressure.
$$ P_B = \frac{B^2}{2\mu_0} $$ magnetic_reconnection_rate magnetic_reconnection_rate_kernel Estimates reconnection rate (Sweet-Parker model simplified).
$$ v_{in} = \frac{v_A}{\sqrt{S}} $$
where $S$ is the Lundquist number. maxwell_gradient Causal wrapper for fields::maxwell_gradient_kernel maxwell_gradient_kernel Calculates the Maxwell gradient (Electromagnetic Field Tensor). orbital_velocity Causal wrapper for mechanics::orbital_velocity_kernel orbital_velocity_kernel Calculates orbital velocity: $v = \sqrt{\frac{GM}{r}}$. partition_function Causal wrapper for stats::partition_function_kernel partition_function_kernel Calculates Partition Function: $Z = \sum e^{-E_i / k_B T}$. poynting_vector Causal wrapper for fields::poynting_vector_kernel poynting_vector_kernel Calculates the Poynting Vector (Energy Flux): $S = E \times B$. proca_equation Causal wrapper for fields::proca_equation_kernel proca_equation_kernel Calculates the Proca Equation (Massive Electromagnetism): $\delta F + m^2 A = J$. quantum_geometric_tensor Wrapper for qgt::quantum_geometric_tensor_kernel quantum_geometric_tensor_kernel Calculates the Quantum Geometric Tensor (QGT) component $Q_{ij}^n(\mathbf{k})$ for band $n$. quasi_qgt Wrapper for qgt::quasi_qgt_kernel quasi_qgt_kernel Calculates the Quasi-QGT $q_{ij}^n(\mathbf{k})$. radioactive_decay Causal wrapper for physics::radioactive_decay_kernel radioactive_decay_kernel Calculates the remaining amount of a radioactive substance: $N(t) = N_0 \cdot 2^{-t / t_{1/2}}$. ray_transfer ray_transfer_kernel Applies an ABCD matrix to a ray vector. relativistic_current relativistic_current_kernel Calculates current density $J^\mu$ compatible with curved spacetime.
$$ J^\mu = \nabla_\nu F^{\mu\nu} $$
(Divergence of Electromagnetic Tensor). resistive_diffusion resistive_diffusion_kernel Calculates the diffusion term of the induction equation.
$$ \frac{\partial \mathbf{B}}{\partial t}_{diff} = \eta \nabla^2 \mathbf{B} $$
On a manifold, this is $-\eta \Delta B$ where $\Delta = d\delta + \delta d$ is the Hodge Laplacian.
Note that $\nabla^2$ in vector calc usually corresponds to $-\Delta$ (negative Laplacian).
The standard diffusion eq is $\partial_t u = + D \nabla^2 u$.
The Hodge Laplacian $\Delta$ is positive definite. So $\nabla^2 \approx -\Delta$.
Thus $\partial_t B = - \eta \Delta B$. rotational_kinetic_energy Causal wrapper for kinematics::rotational_kinetic_energy_kernel rotational_kinetic_energy_kernel Calculates rotational kinetic energy: $K_{rot} = \frac{1}{2} I \omega^2$. schwarzschild_radius Causal wrapper for mechanics::schwarzschild_radius_kernel schwarzschild_radius_kernel Calculates Schwarzschild radius: $r_s = \frac{2GM}{c^2}$. shannon_entropy Causal wrapper for stats::shannon_entropy_kernel shannon_entropy_kernel Calculates Shannon Entropy: $H = -\sum p_i \ln(p_i)$. single_slit_irradiance single_slit_irradiance_kernel Calculates the Single Slit Diffraction Irradiance. snells_law snells_law_kernel Calculates refracted angle using Snell’s Law or returns a Critical Angle error. spacetime_interval Causal wrapper for spacetime::spacetime_interval_kernel spacetime_interval_kernel Calculates the spacetime interval $s^2 = g_{\mu\nu} x^\mu x^\nu$ (squared magnitude in metric). stokes_from_jones stokes_from_jones_kernel Converts a pure state Jones vector to a Stokes vector. thermal_expansion Causal wrapper for mechanics::thermal_expansion_kernel thermal_expansion_kernel Calculates thermal expansion strain: $\epsilon = \alpha \Delta T$. time_dilation_angle Causal wrapper for spacetime::time_dilation_angle_kernel time_dilation_angle_kernel Calculates Time Dilation Angle (Rapidity $\eta$). torque Causal wrapper for kinematics::torque_kernel torque_kernel Calculates torque as the outer product of radius and force: $\tau = r \wedge F$. von_mises_stress Causal wrapper for mechanics::von_mises_stress_kernel von_mises_stress_kernel Calculates Von Mises Stress from a 3x3 Stress Tensor. wave_speed Monadic wrapper for general::wave_speed_kernel wave_speed_kernel Calculates the speed of a wave given its frequency and wavelength. Gate Operator