relativistic particle momentum BD
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Phil's note dated 8.25.08 working in the Bjorken-Drell metric g = diag(1,-1,-1,-1). It defines four-momentum p = m dx/d\u03c4, derives p.p = m^2c^2 and the energy-momentum relation, and recovers the non-relativistic limit mc^2 + p^2/2m. It also covers c=1 units, a caution on covariant derivative index conventions, and hyperbolic boost parameters (cosh u = \u03b3, tanh u = \u03b2, half-angle identities). Some equations were lost in text extraction.
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Relativistic Particle Momentum (BD) PhL 8.25.08
For the moment, let's go with Bjorken Drell's metric which uses g = diag(1,-1,-1,-1).
gμν = gμv = => p.p = gμνpμpν = (p0)2 - p2
where pμ = (p0, p) = (p0, pj) = (p0, - pj). So p means contravariant index (upper).
[ Note: Weinberg use negative of this metric tensor, he uses ημν = -gμν. ] Now, we know that
xμ = (x0, r) = (ct, r) // dim = L => x.x = (ct)2 - r2 x0 = ct
pμ ≡ mdxμ/dτ = γmdxμ/dt = γmvμ where γ = 1/
=> p0 = γmdx0/dt = γmc // dim = ML/T = momentum
p = γm dr/dt = γmv = γβmc = βp0 // dim = ML/T = momentum
p/p0 = β
Then have
p.p = (p0)2- p2 = (γmc)2 - (γmv)2 = (γmc)2 [ 1 - (v/c)2] = (γmc)2 γ-2 = m2c2
Claim: E = γmc2 = cp0 => p0 = E/c // See Goldstein p 202
Then: m2c2 = p.p = (p0)2 - p2 = (E/c)2 - p2
=> (E/c)2 = p2 + m2c2
=> E2 = (cp)2 + (mc2)2 // all terms are dim = E2
There are no 's in special relativity stuff, that is QM.
Non relativistic limit:
E = = mc2 = mc2
= mc2 [ 1 + ½ (p/mc)2 ] = mc2 + p2/2m
Now if we set c=1 the above results become:
p.p = m2 p0 = γm = E γ = 1/
p0 = E E2 = p2 + m2
Warning about : We know that ∂μ = μ = ∂/∂xμ is a covariant vector operator, as shown in physics questions. When we write the bolded 3D symbol, we mean it like this: f(x) = ∂jfj(x). Therefore, we have this unusual situation:
Aμ fμ = A0f0 - Af where Af = Aifi
but
∂μ fμ = ∂0f0 + f where f = ∂jfj
Hyperbolic boost parameters:
γ = ch(u) E = γ mc2 = ch(u) mc2 E/c = ch(u) mc
γβ = sh(u) p = γβmc = sh(u) mc
β = th(u) v = β c = th(u) c
p.p = (E/c)2 - p2 = cosh2 (u) (mc)2 – sinh2 (u) (mc)2 = (mc)2
2sh2(u/2) = [ sh(u) – 1 ] => 2mc2 sh2(u/2) = mc2 [ sh(u) – 1 ] = (E – mc2)
2ch2(u/2) = [ ch(u) + 1 ] => 2mc2 ch2(u/2) = mc2 [ ch(u) + 1 ] = (E + mc2)
sh(u/2) = ch(u/2) = th(u/2) =
sh(u) = p/mc = βγ ch(u) = E/mc2 = γ th(u) = v/c = β = cp/E
And this useful fact
th(u/2) = = sqrt({ (E2 - (mc2)2)/( (E + (mc2))2 } = (cp)/ (E + (mc2))