Dressing vs. renormalization

Question

In the past, my understanding was always that the processes of

• dressing a bare quantity (such as mass or charge) and
• renormalizing these same properties

are different things. Dressing, as far as I had come to learn, was the result of switching on interactions in a formerly free theory and observing that polarization of the vacuum and the emergence of clouds of virtual particles, in particular short-lived particle-antiparticle pairs, would alter physically measurable quantities like mass and charge.

Renormalization, on the other hand, was to me a slightly more formal thing. At its most intuitive, I understood it to be the process of calculating the scale-dependence of couplings, usually with the intent of starting from their values within some microscopic theory $S$ valid at an ultraviolet scale $\Lambda$ and taking their flow down to some lower (usually humanly observable) scale $k$.

Of course, one reason why couplings are scale-dependent in the first place is that, depending on the scale at which a particle's charge or mass is measured, the size of the enclosing cloud of virtual particles and the resulting screening effect will be very different.

However, I recently stumbled upon several text passages that seem to indicate screening and renormalization are actually the same. For instance, Wikipedia in History of quantum field theory states (somewhat vaguely)

What we measure, and hence, what we must take account of with our equations, and what the solutions must account for, are the "renormalized mass" and the "renormalized charge" of a particle. That is to say, the "shifted" or "dressed" values these quantities must have when due systematic care is taken to include all deviations from their "bare values" is dictated by the very nature of quantum fields themselves.

Similarly, Quantum Field Theory by Tom Lancaster and Stephen J. Blundell claims

The process of renormalization may be imagined as a particle dressing itself in interactions.

So is my understanding wrong? Does it lack some important point? Or is it simply that these terms aren't very rigorously defined or that their usage lacks consensus?

Answer 1

A thorough course in renormalisation (like in Collins' book) is really what is necessary to address your concerns, as one should appreciate the big picture of renormalisation as a whole.

Nevertheless, in a nutshell, when we have a free theory, and we add interaction terms, then all of the Green's functions will acquire corrections, as will the parameters of the theory.

The 'dressed' quantity means that the fact it is an interacting theory has been accounted for and quantum corrections are included. For the propagator, we would have,

$$\frac{i}{\gamma^\mu p_\mu - m} \to \frac{i}{\gamma^\mu p_\mu - m -\Sigma(p)}$$

where $\Sigma(p)$ is an infinite sum of a certain class of diagrams. Notice this also has the effect of changing what the physical mass is, since the pole of the propagator has changed.

Renormalisation is a scheme to remove divergences from our computations. Doing so requires adding additional diagrams with counterterms. To generate these involves re-writing the Lagrangian in terms of 'renormalised' quantities plus counterterms.

Note $\mathcal L$ as a whole is exactly the same; it's a change of notation at this point, but it makes a difference in that we treat the entire counterterm Lagrangian, even the mass term, as being an interaction term instead.

In doing so however, the relation between the renormalised parameters and the bare parameters involves a subtraction point or scale $\mu$. The renormalisation group addresses then how the parameters of the theory change with this scale.

Answer 2

According to the theory there is a relation between scale ( renormalization ) and dressed and bare quantities via Gell-Mann and Low's reformulation of Dysons's renormalization program, cf. page 8 of Kerson Huang : A Critical History of Renormalization ( https://arxiv.org/pdf/1310.5533.pdf )