## Expectation value of the position, velocity and acceleration operator, along with the dipole, velocity and acceleration forms of the HHG spectrum, for a 20-cycle circularly polarized pulse at 400 nm and *I* = 10^{14} W cm^{−2} and corresponding HHG spectrum for {
m H}_2^+ at distance *R* = 22 au

**Figure 7.** Expectation value of the position, velocity and acceleration operator, along with the dipole, velocity and acceleration forms of the HHG spectrum, for a 20-cycle circularly polarized pulse at 400 nm and *I* = 10^{14} W cm^{−2} and corresponding HHG spectrum for {\rm H}_2^+ at distance *R* = 22 au.

**Abstract**

Gauge invariance was discovered in the development of classical electromagnetism and was required when the latter was formulated in terms of the scalar and vector potentials. It is now considered to be a fundamental principle of nature, stating that different forms of these potentials yield the same physical description: they describe the same electromagnetic field as long as they are related to each other by gauge transformations. Gauge invariance can also be included into the quantum description of matter interacting with an electromagnetic field by assuming that the wavefunction transforms under a given local unitary transformation. The result of this procedure is a quantum theory describing the coupling of electrons, nuclei and photons. Therefore, it is a very important concept: it is used in almost every field of physics and it has been generalized to describe electroweak and strong interactions in the standard model of particles. A review of quantum mechanical gauge invariance and general unitary transformations is presented for atoms and molecules in interaction with intense short laser pulses, spanning the perturbative to highly nonlinear non-perturbative interaction regimes. Various unitary transformations for a single spinless particle time-dependent Schrödinger equation (TDSE) are shown to correspond to different time-dependent Hamiltonians and wavefunctions. Accuracy of approximation methods involved in solutions of TDSEs such as perturbation theory and popular numerical methods depend on gauge or representation choices which can be more convenient due to faster convergence criteria. We focus on three main representations: length and velocity gauges, in addition to the acceleration form which is not a gauge, to describe perturbative and non-perturbative radiative interactions. Numerical schemes for solving TDSEs in different representations are also discussed. A final brief discussion of these issues for the relativistic time-dependent Dirac equation for future super-intense laser field problems is presented.