In his textbook Quantum Field Theory: A Modern Approach, Michio Kaku first describes quantum field theory as

$\textrm{Quantum field theory} = \left\{\begin{array}{l}\textrm{Group theory} \\ \textrm{Quantum mechanics}\end{array}\right.$

Later on he says,

$\lim_{N\to\infty}\textrm{Quantum Mechanics} = \textrm{Quantum Field Theory}$

where $N$ is the number of degrees of freedom of the system. How compatible are these descriptions?

Consider the group-theoretic description of n-sided polygons given by dihedral groups. A regular polygon with n sides has 2n symmetries, half rotational and half reflective. As $n\to\infty$ one intuitively expects the n-gon to approach a circle and thus would expect that the infinite dihedral group would thus describe the group of symmetries of a circle. But, if I understand this correctly, it doesn’t. In fact the infinite dihedral group is actually an isometry group of $\mathbb{Z}$ which is a line!

Given Kaku’s two definitions, then, I am impelled to ask,

1. is QFT <em>really</em> the limit of QM for as $N \to\infty$; and
2. if it is, then what might we learn by instead considering the limit of large $N$?

Does anyone know if this has been done?

## 15 thoughts on “Between QM and QFT”

1. BlackGriffen says:

Dunno the answers to your questions, but I do have a comment. $\mathbb{Z} \sim \mathbb{Q}$ which we use to approximate $\mathbb{R}$ all the time. Also, calling the isometry group of the n-gons a line isn’t quite accurate. See, the line in this case is compactified at $i = 0 \sim i = n$, where $i$ is the number of times that the elementary rotation is applied.

BG

2. I think that Kaku must mean these to be very heuristic and they certainly cause problems if you take them too seriously.

In particular, there is clearly group theory in nonrelativistic quantum mechanics with a finite number of particles, namely the galilei group and rotational symmetry group. Of course, in QFT we usually use representation theory to determine what we mean by a particle, which is probably what Kaku means. At the level of thinking about field configurations though, there is little difference between the role that group theory plays in QM and QFT.

Regarding the N tends to infinity limit, this should not be thought of as the order of the group going to infinity (the Poincare and Galilei groups both have infinite order in any case). Rather it is about the number of particles going to infinity, from which you can derive the field theoretic limits used in Quantum Statistical Field theory. Another way of thinking about it is to imagine that a “degree of freedom” is represented by a factor in a tensor product decomposition. In a naive approach to quantum field theory, there is one such factor for each spatial location, so there is a continuous infinity of these. In contrast, in nonrelativistic QM for a single particle the Hilbert space is a direct sum of the spaces at each location in space, and you only get another tensor factor when you add another particle.

3. quantummoxie says:

Good to hear from you Matt!

Yeah, I’m sure he meant them somewhat heuristically. I guess I interpreted the “degree of freedom” thing as being the “extent” of the field which, I suppose, would be the same thing as the multiparticle idea.

Sean:

I did not know that’s quite how $Z$ was interpreted in that situation. Whenever I think of $Z$ always think in terms of a line, but that’s probably some naïve holdover from high school or something.

4. Robert Wright says:

I really enjoy your posts, but I have great difficulty reading the mathematics. The combination of ‘thin’ grey font on a black background just doesn’t work

5. quantummoxie says:

Robert:

Thanks! And thanks for the feedback on the math! I’m hoping to get some new software to fix that. It turns out that even if I had a different background, the math apparently would still look thin grey on black from what I can tell. But hopefully the new software will fix that.

6. BlackGriffen says:

Usually for a simple group like the rotation symmetries of a polygon they’ll not just call it $\mathbb{Z}$ but subscript it with the number of elements in the group, eg $\mathbb{Z}_3$ for the rotations of a triangle. Where I’ve seen this usage most is in the relationship between $SU(2)$ and $SO(3)$, ie $SO(3) \sim SU(2) / \mathbb{Z}_2$ because SU(2) is a double covering of SO(3).

7. quantummoxie says:

Yes, I am familiar with the subscript notation (I actually have taught undergraduate group theory and real analysis – scary thought, eh?).

I have a student right now working on a relationship between entropy and permutation symmetry, specifically in terms of the rotational symmetries of n-gons. She’s got a proof that the entropy (which we have associated with the number of symmetries) remains finite as $N \to \infty$.

1. Steve S. says:

How are you defining the entropy in this situation?

1. quantummoxie says:

We’re defining it analogously to the way it is sometimes done in thermodynamics – it’s the logarithm of the number of configurations (microstates) so, in this case, it’s the logarithm of the number of symmetries, i.e. log 2n. I can’t remember what base we’re using.

2. Steve S. says:

Doesn’t log(x) go to infinity as x goes to infinity?

3. quantummoxie says:

She must be doing something else with it. Hmmm…

8. BlackGriffen says:

Thou shalt use base $2.198035052553252 \times 10^{31455787102040073147561}$, and the base shall be $2.198035052553252 \times 10^{31455787102040073147561}$. Only then shall $k_b = \mathrm{J K}^{-1}$. 🙂

1. quantummoxie says:

LOL, too funny.

9. Daniel says:

In my knowledge, QFT incorporates relativistic requirements, this is a requirement independent from the large N limit.

10. quantummoxie says:

Well, I guess I should have made it clearer that I was curious about the difference between relativistic quantum mechanics and quantum field theory. The former includes things like Dirac’s theory of the electron which is not technically a field theory. It serves as the basis for the field theoretic development of QED, but, as Dirac originally developed it, it was merely a relativistic version of quantum mechanics.