# What is 4-momentum: Definition and 39 Discussions

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy E and three-momentum p = (px, py, pz) = γmv, where v is the particle's three-velocity and γ the Lorentz factor, is

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{\displaystyle p=\left(p^{0},p^{1},p^{2},p^{3}\right)=\left({E \over c},p_{x},p_{y},p_{z}\right).}
The quantity mv of above is ordinary non-relativistic momentum of the particle and m its rest mass. The four-momentum is useful in relativistic calculations because it is a Lorentz covariant vector. This means that it is easy to keep track of how it transforms under Lorentz transformations.
The above definition applies under the coordinate convention that x0 = ct. Some authors use the convention x0 = t, which yields a modified definition with p0 = E/c2. It is also possible to define covariant four-momentum pμ where the sign of the energy (or the sign of the three-momentum, depending of the chosen metric signature) is reversed.

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1. ### Expressing Feynman Green's function as a 4-momentum integral

I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do...
2. ### A Exploring the Conservation of 4-Momentum in Quantum Field Theory

why in QFT 4-momentum is conserved? how can it be derived from basic principles of the Hamiltonian formalism? Is it conserved because of the golden rule?
3. ### I Deriving the 4-momentum of a free particle moving in curved spacetime

Consider a free particle with rest mass ##m## moving along a geodesic in some curved spacetime with metric ##g_{\mu\nu}##: $$S=-m\int d\tau=-m\int\Big(\frac{d\tau}{d\lambda}\Big)d\lambda=\int L\ d\lambda$$...
4. ### Squared length of 4-momentum g(p,p)?

Explained in statement.
5. ### B Deriving Energy and Momentum in GR: 4-Component vs. Tensor Approach

In recently closed thread titled Conservation of energy in GR, 4-velocity of popped up baseball along geodestic ##(u^0(r),u^1(r),0,0)## where ##x^1=r,x^2=\theta,x^3=\phi##, is derived. In SR, 4-momentum of ball is ##m(u^0,u^1,0,0)## in contravariant component and ##m(u_0,u_1,0,0)## in covariant...
6. ### I 4-momentum of a massless particle

The 4-momentum of a massless particle traveling in the z direction is (k, 0, 0, k). What is the significance of the value of k? It does not determine the speed since they always travel at light speed. If one particle has momentum (k, 0, 0, k) and another has (j, 0, 0, j) with j not equal to...
7. ### Finding the Energy of an Outgoing Particle in a Momentum Conservation Problem

This problem assumes working in natural units where ##c=1##, and using the Minkowski metric where the time component is positive and the space ones negative (as I know the opposite convention is just as commonly used). EDIT: I had intended to display 4-vectors as bold and the 3-vectors with the...
8. ### Link between 'time' component of 4-momentum and energy

Homework Statement $$E = -\vec{v_{obs}} \cdot \vec{p}$$ Where ## \vec{p} ## is the four momentum, and ## \vec{v_{obs}}## the velocity of the observer. Homework EquationsThe Attempt at a Solution [/B] This was a stated result in a GR course. I look through my SR notes and find that I...
9. ### I Do Fields Affect 4-Momentum Conservation?

When we encounter particle-collision problems that call for invoking the conservation of four-momentum, are we tacitly assuming a field-free idealization (or at least negligible potential energy)? For example, say particles 1 and 2 collide elastically. Then the conservation of four-momentum...
10. ### I Zeroth component of 4-momentum & energy-momentum relation

As I understand it one is forced to use 4-vectors since we require objects that transform as vectors under application of Lorentz transformations and 3-vectors do not (technically they do under rotations, but not under boosts). Equivalenty, if one starts off with Minkowski spacetime from the...
11. ### I Understanding 4-Momentum in Special Relativity

Hello, I am studing elementary particle physics and want to ask something, just to check if I have understood properly. So, as I understand, this is true about four-momentum in special relativity: 1. The square of the sum of particles' four momenta is invariant under Lorentz transformations...
12. ### Electromagnetic field 4-momentum density and inertial frames

Electromagnetic field has a density of energy U = ε/2*E2+ μ/2* H2 And a density of momentum, given by the Poynting vector S = E x H For an element of volume dV you have a four vector of energy and momentum which is [E,P] = dV * [U, S] Being E the energy in the element of volume and P the...
13. ### Mirror 4-Momentum: Find Frequency Change on Reflection

Homework Statement A mirror moves perpendicular to its plane with speed βc. A light ray is incident on the mirror from the “forward” direction (i.e., vm · vl < 0, where vm is the mirror’s 3-velocity and vl is the light ray’s 3-velocity) with incident angle θ (measured with respect to the...
14. ### High Energy Muon scattering - 4-momentum

Homework Statement Explain what the term "four-momentum transfer ##q##" is Show that for a high energy muon scattering at an angle ##\theta##, the value of ##q^2## is given approximately by; ##q^2=2E_iE_f(1-cos(\theta))## where ##E_i## and ##E_f## are the initial and final values of the muon's...
15. ### 4-momentum and relative velocity

Homework Statement In a particular frame of reference a particle with 4-momentum ##P_p## is observed by an observer moving with 4-momentum ##P_o##. Derive an expression for the speed of the particle relative to the observer in terms of the invariant ##P_p.P_o## I am completely stuck on this...
16. ### 3-angular momentum : independent of pivot?

Homework Statement [/B] (i) Prove that dL/dt = 0 (ii) Find the relation between space part and 3-angular momentum vector (iii)Show that 3-angularmomentum vector is independent of pivot Homework EquationsThe Attempt at a Solution [/B] I'm not sure what part (iii) is trying to get at, but I...
17. ### Solve Relativistic Momentum PROBLEM - Extra Factor of u Explained

PROBLEM SOLVED - the worked example I was referring too was wrong :/ ---------------------------- Hello, I've been stuck on a question in one of my SR problem sets for some time now, and managed to find a worked solution to a similar problem online. I've attached an image of the problem (the...
18. ### How can I calculate the speed of the kaon using four-momentum conservation?

Homework Statement So a kaon moving at some speed in the +x direction spontaneously decays into one pion and one anti-pion. The anti-pion moves away with velocity of 0.8c, and the pion moves away with velocity of 0.9c. Mass of kaon = 498 MeV/c^2 Mass of pion/anti-pion = 140 MeV/c^2...
19. ### 4-force, 4-momentum, energy and mass relations.

This is an exercise of Special Relativity the professor asked last week. Sorry for the long post, I hope you don't get bored reading it, also, this is my first post here :shy: Homework Statement Defining the 4-force that acts on a particle as the proper-time variation of the 4-momentum...
20. ### 4-momentum conservation (Deuteron + Photon)

Homework Statement See attachment. Homework Equations The Attempt at a Solution So, \gamma + D \rightarrow p + n (E_1,\underline{k}_1) + (E_2,\underline{0}) = 2(E_3,\underline{k}_3) where I have assumed E3 ≈ E4 and k3 ≈ k4 as mn≈ mp and vp = vn then splitting E and K components up and...
21. ### Checking integration over 4-momentum

Homework Statement I would like to make sure I am performing the integration correctly. It is a loop integral in QFT: \int \frac{d^{4}p}{(2\pi)^{4}}\frac{1}{p^{2}} where p is the 4-momentum, Minkowski space. Homework Equations The Attempt at a Solution I think you must change to...
22. ### Gaussian integrals with 4-momentum

1. Gaussian Please help me calculate some Gaussian integrals in the attached file which are used in my QFT calculations. Thank you very much!
23. ### Gaussian integrals with 4-momentum

I am doing some calculations in QFT. And, in my calculations, I have to deal with 5 Gaussian integrals as followed. Please help me calculate those 5 integrals. Thank you very much!
24. ### Lorentz transformation of y cpmponent for 4-momentum

I have 2 coordinate systems which move along ##x,x'## axis. I have derived a Lorentz transformation for an ##x## component of momentum, which is one part of an 4-momentum vector ##p_\mu##. This is my derivation: \scriptsize \begin{split} p_x &= mv_x \gamma(v_x)\\ p_x &= \frac{m...
25. ### In 4-momentum, why is E the 4th component?

Assuming that c is a "conversion factor" to convert between space and time, Then, in 4-vector, we have x_1 through x_3, and t, where, x/c = t x/c = t, (where t = time, c= lightspeed, x = spatial dimension) If we do what we did to space to get time, to momentum, p/c = m*v/c = m (x/t) / c =...
26. ### Is conservation of the 4-momentum a postulate?

Can the conservation of the 4-momentum be proved, or is it a postulate of special relativity?
27. ### 4-momentum of particles in lab and CM frame

The question: Suppose two identical particles, each with mass m and kinetic energy T, collide head-on. What is the kinetic energy of one in the rest system of the other? The solution: Is given on pg 109 of Griffiths' Introduction to Elementary Particles. Griffiths writes down the total...
28. ### Energy equal to time component of 4-momentum?

I'm sure this question gets brought up a lot, but I can't figure out why this is true. Everywhere I look, people simply equate the two as though it's some axiom, but never an explanation for why. It seems to me like E≠m/√(1-v^2) in general.
29. ### What is the relationship between 4-momentum and energy/momentum components?

Firstly, apologies for the notation. The 4-momentum of a massive particle (rest mass m) is defined by p=mu where u is the 4-velocity. Thus in a frame S in which a particle has 3-velocity u the components of p are [p]=gamma*(mc,mu) How can we then identify the zeroth component of...
30. ### Particle decay using 4-momentum

Homework Statement A particle of mass M, traveling horizontally through the laboratory, decays into two daughter particles, each of mass 0.4M. One of the daughters, A, is produced at rest in the Lab frame. Show that vcm , the speed with which the CM frame moves in the Lab frame, is equal...
31. ### Understanding 4-Momentum: Basics, Interpretation, and Conservation

Basic 4 momentum questions. Trying to understand SR a bit better. Suppose m is non zero for a particle (m being the rest mass of the particle). Then the 4-momentum is related to the 4-velocity by p=mu (in 4 coordinates). The zeroth coordinate of p is therefore m*gamma(v) where v is the...
32. ### Relativity, 4-momentum and 4-velocity

Homework Statement A particle with 4-momentum \textbf{P} is detected by an observer with four-velocity U. Show that the speed, v, of the detected particle in the observers rest frame is given by \sqrt{1-\frac{(\textbf{P}.\textbf{P})c^2}{(\textbf{P}.\textbf{U})^2}} Homework Equations I have a...
33. ### Commutator of 4-momentum and position

Is there a commutation relation between x^{\mu} and \partial^{\nu} if you treat them as operators? I think I will need that to prove this [\$J J^{\mu \nu}, J^{\rho \sigma}] = i (g^{\nu \rho} J^{\mu \sigma} - g^{\mu \rho} J^{\nu \sigma} - g^{\nu \sigma} J^{\mu \rho} + g^{\mu \sigma} J^{\nu...
34. ### What is the 4-momentum of a particle with given rest mass and speed?

Homework Statement A particle has 4-momentum: P^{u} = c(2\sqrt{2} ,1,0,-1) Where c denotes the speed of light. Calculate the particle's rest mass m, its energy E, its speed v and its kinetic energy T. Homework Equations Well the relativistic relation between E, m and p is: m^{2} = E^{2} -...
35. ### How is the 4-momentum 4-vector defined in relativistic QM?

I've been wondering about relativistic quantum mechanics. Elsewhere I'm addressing some comments about this branch of physics but I have never studied it. Is the 4-momentum 4-vector defined in the same way in relativsitic QM or is there a difference? I'm wondering if the time component of...
36. ### Associative Operation with Four Vectors: Is it True?

hi, Is it true that 2*(p+p_1)p_3(p+p_1)p_2 - (p_3)(p_2)(p+p_1)^2 = (p_3)(p_2)(p+p_1)^2 , where p,p_1,p_2,p_3 are four vectors? Or simply: associative this operation?
37. ### Why is 4-momentum usually given as a covector?

I understand the association between vectors and covectors on a Riemannian manifold, but it appears that 4-momentum is given naturally as a covector, instead of vector. 4-position is clearly a vector (for that is the most natural representation of it). Similarly, 4-velocity is also vector...
38. ### Origin of 4-momentum - Hisorical Ref.

I've been dealing with the 4-momentum for too many years without knowing its origin. I'm seeking the physics literature reference to the actual article/text which this item first appeared in relativity. This I hope will give me the person's name, the year and the motivation and actual historical...
39. ### Solving 4-Momentum Problem: Determining Threshold for Triplet Production

I'm having trouble understanding conservation of 4-momentum. My problem is about dertermining the threshold for triplet production from a photon and an electron using 4-momentum conservation. The answer is 4m0c^2. So far, I say the initial energy is: E1^2=(pc)^2 + (m0c^2)^2=(hv)^2+(m0c^2)^2...