summation — from formula anatomySummation is the mathematical operation of adding a sequence of numbers, resulting in their total. In physics, it appears everywhere: from summing forces to find net force, to summing infinitesimal contributions in integrals that describe fields, energies, or probabilities. It is the foundational tool for translating discrete measurements or theoretical components into a single meaningful quantity.
Summation notation (the sigma symbol, Σ) was introduced by Leonhard Euler in the 18th century, though the concept of adding numbers is as old as arithmetic itself.
The operation takes a list of terms (e.g., a1, a2, ..., an) and produces their sum: a1 + a2 + ... + an. In physics, summation is often used with indices to compactly represent large sets of contributions, such as summing over all particles in a system or over all possible states in quantum mechanics. When the terms become infinitesimally small and numerous, summation transitions into integration, but the conceptual core remains the same: accumulation.
Used in virtually every physics calculation, from computing total energy in thermodynamics to summing forces in mechanics and summing over quantum states in statistical physics.
The symbol Σ for summation was chosen by Euler because it is the Greek capital letter for 'S', standing for 'sum'.
| S | variable | Shannon entropy — entropy |
| p_i | variable | Probability of the i-th outcome — probability |
| \sum_i | operator | Sum over all possible outcomes i — summation |
| \ln | operator | Natural logarithm (base e) — natural logarithm |
| H | variable | Hamiltonian operator — hamiltonian joule |
| t | variable | hopping amplitude — energy joule |
| γ_{2j} | variable | Majorana operator at site 2j — majorana operator |
| γ_{2j+1} | variable | Majorana operator at site 2j+1 — majorana operator |
| ∑ | operator | sum over lattice sites j — summation |
| \sigma_f | variable | Combined uncertainty of the function f — uncertainty varies |
| f | variable | Function of multiple measured variables — function varies |
| x_i | variable | i-th measured variable — variable varies |
| \sigma_{x_i} | variable | Uncertainty of the i-th measured variable — uncertainty varies |
| \partial f / \partia | operator | Partial derivative of f with respect to x_i, giving the sensitivity of f to that variable — partial derivative |
| \sqrt{} | operator | Square root of the sum of squared terms — square root |
| \sum_i | operator | Sum over all independent variables i — summation |
| q | variable | entropic index characterizing non-extensivity — entropic index |
| p_i | variable | probability of microstate i — probability |
| W | variable | number of accessible microstates — number of states |
| S_q | variable | Tsallis entropy — entropy joule per kelvin |
| k | constant | Boltzmann constant (or a generalized constant depending on context) — boltzmann constant = 1.380649e-23 joule per kelvin |
| ∑ | operator | sum over all microstates — summation |
| P | variable | Probability distribution over x — probability distribution |
| Q | variable | Probability distribution over x — probability distribution |
| x | variable | Outcome index |
| ∑ | operator | Summation over the discrete sample space — summation |
| |·| | operator | Takes the non-negative magnitude — absolute value |
| \mathcal{L} | variable | Cross-entropy loss value — loss |
| y_i | variable | True probability (or one-hot label) for class i — probability |
| p_i | variable | Predicted probability for class i — probability |
| \sum_i | operator | Adds the terms for each class — summation |
| \log | operator | Computes the logarithm to base e — natural logarithm |