We consider rectilinear free-space propagation of electromagnetic wavepackets using electromagnetic field theory, scalar wavepacket propagation, and quantum-mechanical formalism. We demonstrate that spatially localized wavepackets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals $c^2$. These velocities are also known as the 'energy' and 'momentum' velocities, introduced by K. Milton and J. Schwinger. We illustrate general conclusions by explicit calculations for Gaussian beams and wavepackets, and also highlight subtleties of the quantum-mechanical description based on the 'photon wavefunction'.
Subluminal and superluminal velocities in the propagation of light have intrigued scientists for several decades 1 .
Multiple distinct velocities characterize light propagation, such as phase, group, signal, and energy-transfer velocities. Furthermore, various physical mechanisms can lead to subluminal or superluminal behavior: interactions with matter [2][3][4][5][6] , nontrivial quantum-vacuum effects 7 , local phase gradients in structured free-space light 8-10 , specially engineered space-time wavepackets [11][12][13] , and the tilt of plane-wave Fourier components in transversely confined light beams [14][15][16][17][18][19][20][21][22][23] .
In this work, we address the simplest scenario among these, namely, the free-space propagation of electromagnetic wavepackets without engineered internal structure. The only requirement is suitable spatial confinement of the wavepacket, which is necessarily entails a corresponding spread in momentum (wavevector) space. As theoretically described and experimentally observed in 17,18,20,23 , such wavepackets or single photons propagate with a subluminal group velocity v g < c, determined by the transverse confinement of the wavepacket. In addition, we show that the wavepacket can also be characterized by a superluminal phase velocity v ph > c, such that v g v ph = c 2 , see Fig. 1.
We trace the origin of this result through several complementary approaches and establish useful connections between relativistic field theory, quantum mechanics, and previous studies of this phenomenon.
We begin with a textbook exposition of electromagnetism based on relativistic field theory [24][25][26] . It is known that an electromagnetic field can be characterized by its energy density U and momentum density P (proportional to the Poynting vector). For a localized wave packet propagating in free space, the spatial integrals of these quantities are conserved. According to Noether’s theorem, the conservation of energy and momentum follows from invariance under time and space translations, respectively. In addition, there are conservation laws for angular momentum and the so-called ‘boost momentum’, associated with invariance under spatial rotations and and spatio-temporal rotations (i.e., Lorentz boosts).
Wave-packet propagation is closely related to boostmomentum conservation 23,26 , which can be written as
From here, one readily finds that the energy centroid,
, obeys the equation of motion
where we have used the fact that the total energy U, d 3 r is a conserved time-independent quantity. Equation (2) represents the field-theoretical analogue of the relativistic equation of motion for a point particle,: v = c 2 p/E 27 . Importantly, the energy-centroid velocity (2) is subluminal for localized wavepackets. This can be seen directly from the explicit expressions for the electromagnetic energy and momentum densities: U = (E 2 + H 2 )/2 and P = (E × H)/c, where E and H are the electric and magnetic fields, and we use Gaussian units omitting inessential numerical constants. Obviously, c|P| ≤ U , and equality is attained for null fields arXiv:2602.17576v1 [quant-ph] 19 Feb 2026
, where equality is possible only for null fields with a fixed direction of P. This means that |v E | = c only for plane waves, which are spatially delocalized.
By analogy with ’energy velocity’ (2), Milton and Schwinger introduced the concept of a ‘momentum velocity’ in their textbook 26 . Using an electromagnetic analogue of the virial theorem, they derived the relation
This motivated the definition of the ‘momentum centroid’ projected onto the integral momentum direction: R P • P d 3 r = r • P d 3 r. Taking the time derivative, and using Eq. ( 3) and the conservation of the total momentum P d 3 r, we find that this centroid propagates with a velocity v P satisfying
Comparing Eqs. ( 2) and ( 4), we find that v P • v E = c 2 . Consequently, the longitudinal (i.e., along the total momentum) component of the ‘momentum velocity’ is superluminal for localized wavepackets:
Remarkably, Milton and Schwinger concluded from Eqs. ( 1)-( 4) that 26 : “If the flow of energy and momentum takes place in a single direction, it would be reasonable to expect that these mechanical properties are being transported with a common velocity v E = v P = v, |v| = c, …, which results express the mechanical properties of a localized electromagnetic pulse carrying both energy and momentum at the speed of light, in the direction of the momentum”. However, this argument contains an intrinsic inconsistency: in a localized electromagnetic wavepacket the flow of energy does not take place in a single direction. Indeed, a wave packet arises from the interference of multiple plane waves propagating in different directions; it diffracts during propagation, and energy flows in both longitudinal and transverse directions 28 . (Note that nondiffracting Bessel beams 29 are not square-integrable and therefore do not represent properly loc
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