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

Mass renormalization actually has classical underpinning, going back to the 19th century, and what you see in quantum field theory is a quantized form of it. It is not something introduced by quantum theory, but inherited by quantum theory from classical physics.

As for field and charge renormalization: the question you should be asking is now "why" but "what" is being renormalized? The "bare" versus "dressed" distinction for charges and fields are those taken with respect to them referred to in Gaussian units. This unfortunate choice of units, which today survives only in the theoretical literature as one of its few remaining hold-outs, leads to confusion, false attribution: this very issue being a case in point.

This is a follow-up on the comments I made in my reply here Gaussian Unit Of Charge And Force

For charges: a Gaussian charge, which we'll denote $q$ - as in the reply - is related to ordinary charge, which we'll denote $e$, as measured (say) in Coulombs by $q = e/\sqrt{4πε_0}$, where $ε_0$ is the permittivity of the vacuum.

When we associate a renormalization coefficient $\sqrt{Z_3}$ with $q$, what we're actually doing is renormalizing $ε_0$ by $1/Z_3$. It's not the charge that has the "bare" versus "dressed" distinction and is not the charge that's being renormalized, but the permittivity and the vacuum, itself!

For the electromagnetic field, denoting the Gaussian version of the potential by $a$ and the Si version by $A$, the relation between the two is $a = \sqrt{4πε_0}c A$. So, when we associate a renormalization coefficient - the same one (in fact) - $\sqrt{Z_3}$ with $a$, what we're actually doing is associating $1/Z_3$ with $ε_0$ and renormalizing it.

In both cases, it's the permittivity that's being renormalized, not the fields or charges. The use of Gaussian units obscures this and confuses the issue.

An analogous situation occurs with other gauge fields. To see what the analogue is, write out the Maxwell-Lorentz Lagrangian density in ordinary, SI, units: $$_{ML} = -¼ ε_0 c \sqrt{|g|} g^{μρ} g^{νσ} F_{μν} F_{ρσ}.$$ By comparison, the Yang-Mills Lagrangian density has the form: $$_{YM} = -¼ k_{ab} \sqrt{|g|} g^{μρ} g^{νσ} F^a_{μν} F^b_{ρσ},$$ where the indices $a,b,...$ are taken with respect to a basis of the Lie algebra associated with the gauge field (e.g. color $SU(3)$ for the QCD field), and $k_{ab}$ is the adjoint-invariant metric associated with the gauge group and its Lie algebra. It is usually (but not always) suppressed in the theoretical literature which - correspondingly - plays fast and loose with its careless handling of the indices failing to distinguish covariant from contravariant, or even recognize that there is a difference! For simple gauge groups, like $SU(3)$, an adjoint invariant metric is determined uniquely, up to scale, as a multiple of the Killing metric, $κ_{ab} = \sum_{0≤c,d<n} f^c_{ad} f^d_{bc}$, where the $f^c_{ab}$ are the structure coefficients associated with the Lie algebra basis $\left(Y_a: 0 ≤ a < n\right)$, i.e. the coefficients in the Lie bracket relations $\left[Y_a, Y_b\right] = \sum_{0≤c<n} f^c_{ab} Y_c$. (For $SU(3)$, $n = 8$.) The extra coefficient is the "coupling coefficient" $g$, itself, with $k = κ/g^2$ up to signs and factors (I forget whether there's a $4π$ or not) and is frequently, in the theoretical physics literature, migrated over into the fields as part of their definition by rescaling them in much the same way that the Gaussian $a$ is rescaled relative to SI $A$.

The role analogous to the vacuum permittivity $ε_0$ in $ε_0 c$ is played by $k_{ab}$ and, by extension, by its coefficient $g$. So, we could consider $k_{ab}/c$ as the gauge version of "vacuum permittivity". The role played by $g$ is thus analogous to $1/\sqrt{4πε_0}$.

So, when we're talking about the "running of the couplings", once again, what we're actually referring to is the renormalization of the vacuum and the running of its coefficients: here, the metric $k_{ab}$ and its coefficient $g$.

For semi-simple gauge groups, and products of them with Abelian gauge groups, the chief example in mind being $SU(3)⊗SU(2)⊗U(1)$, the metric is a direct sum of the metrics associated with each factor. So, you have several metric independent coefficients in place of just one: here $g_s$ for color $SU(3)$, $g$ for isospin $SU(2)$ and $g'$ for hypercharge $U(1)$.

Answer 3

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 )