Shall we skip the step of explicit regularization in the process of renormalization?
Answers
Answered by
0
In the process of renormalization, regularization is usually cited as indispensable in taming infinities encountered in quantum field theory. Is explicit regularization really necessary?
Let's take for example the fermion propagator
G=i/p−m0−Σ(/p)+iϵ=i(1−b(p2))/p−(m0+a(p2))+iϵ,G=i⧸p−m0−Σ(⧸p)+iϵ=i(1−b(p2))⧸p−(m0+a(p2))+iϵ,
where self energy is expressed as
Σ(p)=a(p2)+b(p2)/p.Σ(p)=a(p2)+b(p2)⧸p.
The propagator has a pole at
mp=m0+a(m2p)1−b(m2p),mp=m0+a(mp2)1−b(mp2),
where m0m0 is bare mass (infinite) and mpmp is the physical mass (finite).
One may rearrange the above Fermion propagator via introducing modified self energy Σ^(/p)Σ^(⧸p) so that
G=iZ/p−mp−Σ^(/p)+iϵ,G=iZ⧸p−mp−Σ^(⧸p)+iϵ,
where Σ^(/p)Σ^(⧸p) is defined as
Z−1Σ^(/p)=[a(p2)−a(m2p)]+[b(p2)−b(m2p)]/p,Z−1Σ^(⧸p)=[a(p2)−a(mp2)]+[b(p2)−b(mp2)]⧸p,
and
Z=11−b(m2p).Z=11−b(mp2).
Note that the difference a(p2)−a(m2p)a(p2)−a(mp2) is finite, even though a(p2)a(p2) and a(m2p)a(mp2) are individually infinite. If we follow the regime of sticking with finite differences (i.e. a(p2)−a(m2p)a(p2)−a(mp2)) and measurable quantities (i.e. mpmp) only, then the explicit regularization schemes (such as the widely used dimensional regularization) are not needed at all.
An added note on the difference between two infinite quantities. Take the following example,
∫r01xdx−∫r001xdx=∫rr01xdx=ln(rr0).∫0r1xdx−∫0r01xdx=∫r0r1xdx=ln(rr0).
Hard core mathematicians will be leery of the first step and demand some form of regularization. Do physicists, while not fazed by the lack of mathematical rigor with things like path integral, really need a formal explicit regularization to arrive at the final result?
One may call the above procedure implicit regularization. Similar idea has already been picked up by some researchers (search for the key word "implicit regularization"), though in a different fashion as framed here. The merit of implicit regularization is that it circumvents various pitfalls besieging explicit regularization, e.g. violation of gauge invariance in cutoff regularization or the γ5γ5 issue in dimensional regularization.
Let's take for example the fermion propagator
G=i/p−m0−Σ(/p)+iϵ=i(1−b(p2))/p−(m0+a(p2))+iϵ,G=i⧸p−m0−Σ(⧸p)+iϵ=i(1−b(p2))⧸p−(m0+a(p2))+iϵ,
where self energy is expressed as
Σ(p)=a(p2)+b(p2)/p.Σ(p)=a(p2)+b(p2)⧸p.
The propagator has a pole at
mp=m0+a(m2p)1−b(m2p),mp=m0+a(mp2)1−b(mp2),
where m0m0 is bare mass (infinite) and mpmp is the physical mass (finite).
One may rearrange the above Fermion propagator via introducing modified self energy Σ^(/p)Σ^(⧸p) so that
G=iZ/p−mp−Σ^(/p)+iϵ,G=iZ⧸p−mp−Σ^(⧸p)+iϵ,
where Σ^(/p)Σ^(⧸p) is defined as
Z−1Σ^(/p)=[a(p2)−a(m2p)]+[b(p2)−b(m2p)]/p,Z−1Σ^(⧸p)=[a(p2)−a(mp2)]+[b(p2)−b(mp2)]⧸p,
and
Z=11−b(m2p).Z=11−b(mp2).
Note that the difference a(p2)−a(m2p)a(p2)−a(mp2) is finite, even though a(p2)a(p2) and a(m2p)a(mp2) are individually infinite. If we follow the regime of sticking with finite differences (i.e. a(p2)−a(m2p)a(p2)−a(mp2)) and measurable quantities (i.e. mpmp) only, then the explicit regularization schemes (such as the widely used dimensional regularization) are not needed at all.
An added note on the difference between two infinite quantities. Take the following example,
∫r01xdx−∫r001xdx=∫rr01xdx=ln(rr0).∫0r1xdx−∫0r01xdx=∫r0r1xdx=ln(rr0).
Hard core mathematicians will be leery of the first step and demand some form of regularization. Do physicists, while not fazed by the lack of mathematical rigor with things like path integral, really need a formal explicit regularization to arrive at the final result?
One may call the above procedure implicit regularization. Similar idea has already been picked up by some researchers (search for the key word "implicit regularization"), though in a different fashion as framed here. The merit of implicit regularization is that it circumvents various pitfalls besieging explicit regularization, e.g. violation of gauge invariance in cutoff regularization or the γ5γ5 issue in dimensional regularization.
Answered by
0
Implicit regularisation works fine for scalar and spinor theories, but it fails for gauge theories. I invite you to compute the photon self-energy without introducing an explicit (and gauge-invariant) regulator, and check whether the Ward identity is satisfied.
Similar questions