4-Vector Momentum
The previous discussion was an attempt to outline some of the basics of classical kinematics, which we will need to extend into the realm of relativistic particle collisions. In this discussion, the goal is to simply outline how the concept of 4-momentum develops out of the relativistic energy equation:
[1]
While there is much in Feynman’s QED model that appears to be abstracted from our intuitive understanding of physical reality, the previous discussion of the collision cross-section and classical kinematics might have raised some questions as to how much of a quantum electron-positron collision can still be physically observed as opposed to mathematically inferred. In the diagram below, we might define the electron-positron (p1,p2) pair being associated with an initial state that can be observed, such that the electron-positron (p3,p4) pair on the right might be aligned to some final state, which can also be observed or, at least, measured in some way.
The diagram above presents no specific details as to the nature of the collision within the yellow zone, but it is an overall requirement that the initial and final observable states must comply with the conservation of energy and momentum, as well as charge. In fact, according to Feynman’s rules, the laws of conservation must be adhered to at all vertex coupling points, e.g. as the point of intersection in the following annihilation example.
Therefore, following on from the previous discussion of classical kinematics, it might be useful to establish an initial descriptive model, which is essentially classical, so that the conservation of energy may be considered as a scalar quantity, while the conservation of momenta has to be considered as a vector quantity. As such, the inbound positron and electron particles are assigned energy values (E_{1}) and (E_{2}) and momenta (p_{1}) and (p_{2}) respectively. As a scalar quantity, the energy of the inbound positron and electron is combined and assumed to be converted into the virtual photon during annihilation, i.e.
[2]
However, at this point, it might be sensible to highlight that ‘virtual photons’ do not necessarily conform to the description of ‘real photons’ , when it comes to the conservation of energy. Likewise, we might consider the conservation of momentum, for the annihilation diagram above, in terms of a 2D model such that the combined momentum of the positron and electron would be the sum of the [x,y] components of (p_{1}) and (p_{2}). Examination of the specific annihilation diagram above might suggest that the y-components of (p_{1}) and (p_{2}) might cancel, leaving only the x-components of momentum being conserved in the photon. If so, vector geometry would imply:
[3]
While the suggestion of an analogous billiard ball collision might be inappropriate, in the current context, it is only intended as a reference model for some of the ideas that still seem to apply, if the conservation of energy and momentum have to be enforced at all collision vertices within a Feynman diagram.
However, as already highlighted, Feynman diagrams should not be taken as a literal representation of spacetime, where in the case of annihilation the virtual photon may appear to only exist in time and then only in space in the two diagram above. However, this is said to be a consequence of the diagrams being drawn in momentum space and either diagram could have been drawn with the photon at an angle, which would have then been more suggestive of some sort of physical propagation in both time and space. In practice, the only difference between the ‘annihilation’ and ‘scattering’ is in the definition of the initial and final states. As such, the differences in the internal interaction is only reflected in how the 4-momentum of the virtual photon is defined in terms of the external momenta of the particles. However, it may still not be clear as to how energy-momentum is being transferred to and by the virtual particle, i.e.
Is this a particle interaction or some form of quantum wave superposition?
Of course, such questions go to the core of QFT, which ultimately has to be quantify in terms of quantum mechanics, perturbation theory, electrodynamics and special relativity plus a lot of abstracted and sophisticated mathematics. However, from the perspective of trying to gain some general understanding of the physics, it might be worth some initial review of the idea of 4-vectors before further considering some the mathematical formalism linked to this subject:
[4]
Note: For the purposes of this initial introduction, 4-vectors will be denoted with an overstrike, while a 3-vector will be highlighted in bold. |
As indicated, QFT is predicated on combining quantum mechanics and special relativity, such that we might initially describe 4-position in terms of a Minkowski spacetime diagram:
[5]
Here (x_{0},x_{0}’) are the generalisation of time (ct) in two different frames of reference, while (x_{1},x_{1}’) represents a 1-dimensional separation in space, such that the spacetime interval [s] or proper time [τ] are invariant in all frames of reference. At this point, we might simply highlight that the invariance of a spacetime interval [s,τ] is linked to the Lorentz factor [γ]. On this basis, we might now extend the 4-vector concept to velocity and momentum:
[6]
However, in the case of 4-momentum, we might re-arrange the definition of the scalar component (p_{0}), such that it is linked to energy (E):
[7]
As such, we might substitute the result in [7] back into [6]:
[8]
We might now interpret [8] in two ways. First, by analogy to a Minkowski spacetime diagram, as referenced in [5], which was shown to lead to an invariant quantity called the spacetime interval. However, there is an implication in [8] that if we plot energy against momentum, we might end up with another invariant quantity, although this might be best explained in terms of 4-momentum, which is constrained to 1-dimension:
[9]
In [9], we see the encapsulation of the relativistic energy [E] equation being sourced from the definition of 4-momentum, where rest mass energy is the invariant quantity in all frames of reference. However, what may not be obvious from the form of [9] is that this is an expression of the relativistic energy of just one particle, which has an energy associated with (p_{0}) and a 1-dimensional momentum (p_{1}) that is analogous to the rule encapsulated in a Minkowski spacetime diagram, as per [5]. However, we might clarify this situation in terms of two particles using the notation [p_{1}=u] and [p_{2}=v]
[10]
In the context of [10], the sum of [ p_{u}+p_{v} ] represents the addition of two 3-dimensional vectors in space, such that the result of the annihilation collision aligns to the form in [3]:
[11]
The angle [θ] is the sum of the incident angle of the particles [u,v] with respect to the horizontal in the annihilation diagram, which if [45^{o}+45^{o}=90^{o}] reduces the last term to zero. Based on [9], we might give the invariant quantity (m_{0}c)^{2} the label [s], which will later be discussed in terms of 1 of 3 Mandelstam variables.
[12]
By way of cross reference to standard text, which use natural units, the equation in [12] is often reduced to the following form:
[13]
However, the effects of relativity causes both kinetic energy and momentum to increase in comparison to the rest mass energy, such that the latter might become negligible in comparison to the other terms in [12]:
[14]
While a ‘real’ photon would defined its energy and momentum in terms of [E=hf] and [p=E/c], a virtual photon has some notional concept of rest mass energy, as defined by [s]. So, by way of a summary:
A virtual particle is said to be ‘off-shell’ because it does not comply with the relativistic equation in [9], because virtual particles, such as a photon can have (+/-) mass. This type of mass gives rise to the terminology of an ‘off-mass shell’, which might be better described as ‘borrowed energy’ within the confines of the energy/time variant of the uncertainty principle. So while the conservation of energy and momentum must apply at the vertex of a Feynman diagram, the virtual photon associated with either annihilation or scattering can assume an off-shell mass/energy, while all observed particle remain on-shell. |