Derivation of the energy-momentum relation
Shan Gao∗
October 18, 2010
Abstract
It is shown that the energy-momentum relation can be simply determined
by the requirements of spacetime translation invariance and relativistic
invariance.
Momentum and energy are two of the most important concepts of modern
physics. Their relation has been widely used in Newtonian mechanics and quantum mechanics in an approximate form, as well as in relativistic mechanics and
quantum field theory in an exact form. Yet, as far as we know, a fundamental
derivation of this pivotal relation seems still missing in the literature. Most
“derivations” are based on the somewhat complex analysis of an elastic collision
process. Moreover, they resort to either some Newtonian limit (e.g. p = mv)
or some less fundamental relation (e.g. p = Eu/c2 ) or even some mathematical
intuition (e.g. four-vectors) [1-6]. As we think, the logic of these “derivations”
seems a little upside-down, and they are only heuristic demonstrations of the
energy-momentum relation. So simple and fundamental relation the energymomentum relation is, and thus its derivation should be simpler, based on more
fundamental postulates. In this short note, we will present such a derivation,
based only on spacetime translation invariance and relativistic invariance.
In order to derive the energy-momentum relation we must start from the
quantum origin of energy and momentum, the momentum eigenstate ei(px−Et) .
It is well known that the momentum operator P and energy opertaor H are
defined as the generators of space translation and time translation, respectively.
The evolution law of an isolated system satisfies spacetime translation invariance due to the homogeneity of space and time. Time translational invariance
requires that H have no time dependence, namely dH/dt = 0 (see, e.g. [11],
p.295). Space translational invariance further requires that the generators of
space translation and time translation are commutative, namely [P, H] = 0
(see, e.g. [11], p.293). Therefore, P and H have common eigenstates, which
∗ Unit for HPS and Centre for Time, University of Sydney, NSW 2006, Australia. E-mail:
[email protected].
1
turn out to assume the form ei(px−Et) , where p and E are the corresponding
eigenvalues. Since H, the generator of time translation, determines the evolution, these eigenstates are also the solutions of the free evolution equation, and
they can be the actual states of an isolated system (e.g. a free particle) with
definite momentum and energy.
Now we will derive the relation between momentum p and energy E in
relativistic domain. Consider two inertial frames S0 and S with coordinates
x0 , t0 and x, t. S0 is moving with velocity v relative to S. Then x, t and x0 , t0
satisfy the Lorentz transformations:
x0 = p
x − vt
(1)
1 − v 2 /c2
t − xv/c2
t0 = p
1 − v 2 /c2
(2)
Suppose the state of a free particle is ψ = ei(p0 x0 −E0 t0 ) , an eigenstate of P , in
S0 , where p0 , E0 is the momentum and energy of the particle in S0 , respectively.
When described in S by coordinates x, t, the state is
ψ=e
i(p0 √
x−vt
1−v 2 /c2
t−xv/c2
1−v 2 /c2
−E0 √
)
2
=e
p0 +E0 v/c
E +p v
i( √
x− √ 0 20 2 t)
2 2
1−v /c
1−v /c
(3)
This means that in frame S the state is still the eigenstate of P , and the corresponding momentum p and energy E is1
p0 + E0 v/c2
p= p
1 − v 2 /c2
(4)
E0 + p0 v
E=p
1 − v 2 /c2
(5)
We further suppose that the particle is at rest in frame S0 . Then the velocity
of the particle is v in frame S 2 . Considering that the velocity of a particle in the
momentum eigenstate ei(px−Et) or a wavepacket superposed by these eigenstates
is defined as the group velocity of the wavepacket, namely
u=
dE
,
dp
(6)
we have
dE0 /dp0 = 0
(7)
dE/dp = v
(8)
1 Alternatively we can also obtain the transformation of momentum and energy by directly
requiring the relativistic invariance of momentum eigenstates ei(px−Et) , which leads to the
relation px − Et = p0 x0 − E0 t0 .
2 Note that we can also get this result from the definition Eq. (6) by using the above
transformation of momentum and energy Eq.(4) and Eq.(5).
2
The first equation Eq.(7) means that E0 and p0 are independent. Moreover,
since the particle is at rest in S, E0 and p0 do not depend on v. By differentiating
both sides of Eq.(4) and Eq.(5) relative to v we obtain
v p0 + E0 v/c2
dp
E0 /c2
= 2
3 +
1
2
2
dv
c (1 − v /c ) 2
(1 − v 2 /c2 ) 2
(9)
v E0 + p0 v
p0
dE
= 2
+
1
dv
c (1 − v 2 /c2 ) 23
(1 − v 2 /c2 ) 2
(10)
Dividing Eq.(10) by Eq.(9) and using Eq.(8) we obtain
p0
p
1 − v 2 /c2
=0
(11)
This means p0 = 0. Inputing this important result to Eq.(5) and Eq.(4), we
immediately have
E=p
E0
,
(12)
E0 v/c2
p= p
,
1 − v 2 /c2
(13)
1 − v 2 /c2
Then the energy-momentum relation is:
E 2 = p2 c2 + E02
(14)
where E0 is the energy of the particle at rest, called rest energy of the particle,
and p and E is the momemtum and energy of the particle with velocity v.
By defining m = E/c2 as the mass of the particle, we can further obtain the
familar relation p = mv, where m satisfies
m= p
m0
1 − v 2 /c2
(15)
where m0 is usually called the rest mass of the particle. In nonrelativistic domain, the energy-momentum relation reduces to E = p2 /2m0 3 , and momentum
is p = m0 v and energy is E = 12 m0 v 2 . Note that the famous Einstein equation
E = mc2 is only a convenient definition from a more fundamental view, and we
can in principle avoid talking about mass in modern physics (cf. [7-10]).
To sum up, we have shown that the requirements of spacetime translation invariance and relativistic invariance can naturally lead to the relativistic energymomentum relation.
3 Once we have obtained this energy-momentum relation, we can further derive the
Schrödinger equation in quantum mechnics. A detailed discussion will be given in another
paper.
3
References
[1] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on
Physics, vol. I (Reading, Addison-Wesley, 1963), pp. 16/6-16/7.
[2] E. F. Taylor and J. A. Wheeler, Spacetime Physics (New York, Freeman,
1966), pp. 106- 108.
[3] R. C. Tolman, Relativity, Thermodynamics, and Cosmology (Oxford, Clarendon Press, 1934). Reissued by Dover (1987), pp. 43-44.
[4] N. D. Mermin, Space and Time in Special Relativity (New York, McGrawHill, 1968). Reissued by Waveland Press (1989), pp. 207-214.
[5] G. F. R. Ellis and R. M.Williams, Flat and Curved Space-Times (Oxford,
Oxford University Press, 2000), pp. 330-331.
[6] S. Sonego and M. Pin, Deriving relativistic momentum and energy. Eur.
J. Phys. 26, 33-45 (2005).
[7] C. G. Adler, Does mass really depend on velocity, dad? Am. J. Phys. 55,
739-743 (1987).
[8] L. B. Okun, The concept of mass. Phys. Tod. 42 (6), 31-36 (1989);
[9] T. R. Sandin, In defense of relativistic mass, Am. J. Phys. 59 (11): 1032
(1991).
[10] L. B. Okun, Mass versus relativistic and rest masses. Am. J. Phys. 77
(5), 430 (2009).
[11] R. Shankar, Principles of Quantum Mechanics, 2nd ed. (New York, Plenum,
1994).
4