Quantum Mechanics Invalidates Naive Dimensional Analysis

Anomalous dimensions are counterintuitive for beginners, important, together with omnipresent

I take away maintain mentioned this subtle surprising lawsuit of quantum mechanics inward the blog send service against the Lagrangians ii days ago. Too many people retrieve that the quantization only way to add together elegant hats inward a higher house all the degrees of liberty together with increase the frequency at which they say words similar "weird" or "entanglement". But they all the same think classically.

That's non what you lot take away maintain needed to practise to retrieve scientifically later 1925. Quantum mechanics is a fundamentally unlike theory whose framework is rattling unlike conceptually together with whose quantitative predictions may sometimes move similar equally inward classical physics but sometimes they tin give the sack move unlike or totally different. And fifty-fifty when quantum mechanics ends upwardly with the same or similar conclusions equally classical physics, the derivation or argumentation may move rattling different. One of the omnipresent changes that quantum mechanics forces upon us is a new, modified dimensional analysis inward plain theory. What's going on?

In classical plain theory, you lot take away maintain the activity \(S=\int d^D x\,{\mathcal L}\) if 1 exists at all. Because the activity is dimensionless (in SI units, it has the dimension of the activity i.e. the same equally \(\hbar\) but I volition laid \(\hbar=1\)) – exponents e.g. inward the Feynman integrand \(\exp(iS/\hbar)\) take away maintain to move dimensionless – the Lagrangian density \({\mathcal L}\) has the dimension of \(M^D\), the volume to the mightiness of the spacetime dimension. The volume has the dimensions of \([M]=1/[L]\) inward the \(c=\hbar=1\) units.




Because nosotros oft write the bosonic kinetic damage inward the Lagrangian density as\[

{\mathcal L}_{\rm kin} = \frac 12 \partial_\mu \Phi \partial^\mu \Phi

\] i.e. without whatever adjustable coefficients – nosotros may normalize \(\Phi\) together with so that the coefficient isn't in that location or it is just \(1/2\) etc. – nosotros may meet that inward \(D\) spacetime dimensions, \(\partial_\mu\) has the dimension of volume together with so \(\Phi\) has to take away maintain the dimension \(M^{(D-2)/2}\) for the Lagrangian density to take away maintain units of \(M^D\). For example, inward the \(D=4\) spacetime that many readers incorrectly believe to inhabit, \(\Phi\) – bosonic fields and/or gauge potentials – take away maintain the dimension of volume \(M\).




Naively, a beginner is rattling probable to believe that this derivation of the units of the Klein-Gordon plain \(\Phi\) has to move valid inward quantum mechanics, too. After all, the lid doesn't modify anything virtually the units, does it? Well, the lid doesn't modify the units straight but the human relationship betwixt the degrees of liberty – which larn formidable operators thank you lot to their elegant hats – profoundly transforms (they don't commute with each other, with other things) which is why the dimensional analysis inward a higher house is ultimately corrected – together with may move completely wrong – inward quantum mechanics.

The actual units of the operator are \(M^\Delta\) where \(\Delta = \Delta_0+\gamma(g)\) where \(\gamma(g)\) is the anomalous dimension. Whatever is the difference betwixt the actual dimension \(\Delta\) together with the naively, classically derived dimension \(\Delta_0\) is called the anomalous dimension. It may depend on the gauge couplings together with similar parameters inward the plain theory.

What is the simplest, canonical example? Textbooks of quantum plain theory may alternative some complicated examples to start with. But I believe the exponential operator inward \(D=2\) is the best, solvable, pretty, introductory example. What is the theory together with what is the operator? Just consider a Klein-Gordon plain \(X\) inward the \(D=2\) dimensional spacetime – well, let's telephone outcry upwardly it the globe canvas (the history of a propagating string inward string theory). The Klein-Gordon activity (the kinetic term) is\[

southward = \frac 12 \int d^2\sigma\, \partial_\alpha X \partial^\alpha X

\] It's a normal Klein-Gordon action. Because nosotros desire to telephone outcry upwardly this realm "world sheet" together with non "spacetime", I chose a novel call for coordinates, \(\sigma^\alpha\) instead of \(x^\mu\), together with the indices were taken from the start of the Greek alphabet. Great. On the globe sheet, you lot may all the same expand the quantum plain \(X(\sigma^0,\sigma^1)\) inward damage of creation together with annihilation operators multiplied past times bird waves, just similar you lot practise inward the \(D=4\) Klein-Gordon theory.

You're used to the calculation of correlation functions of operators that are polynomial inward \(X\), right? But inward \(D=2\), nosotros take away maintain something novel together with interesting. What are the units of \(X\)? In \(D\) dimensions, it's "naively" \(M^{(D-2)/2}\), equally I derived previously. As you lot tin give the sack see, for \(D=2\), it's zero: the plain \(X\) is dimensionless; the ii derivatives \(\partial_\alpha\) already take away maintain the right units of volume to cancel the two-dimensional integral over the globe canvas coordinates. For this reason, you lot may house it inward the exponent together with consider the operator\[

{\mathcal O} = \exp(ikX)

\] where \(k\) is some coefficient. Note that \(ikX\) must move dimensionless together with because \(X\) has no mightiness of "length on the globe sheet", \(k\) doesn't take away maintain one, either. Both of them take away maintain some units involving the "length inward the spacetime" but the spacetime is a unlike infinite than the globe canvas where this theory is defined – inward effect, it's some internal "configuration space", together with so these spacetime units are dimensionless from the globe canvas viewpoint. (To allow \(X\) to move written inward normal "spacetime units", some coefficients that depend on \(\alpha'\), which has the units of squared length, should move added everywhere but I volition omit them.)

Just similar the exponent within \(\exp(\dots)\) has to move dimensionless – e.g. because the Taylor expansion for the exponential adds the exponent together with its powers to \(1\) together with so all of those damage take away maintain to take away maintain the same units together with move dimensionless, just similar \(1\) – the exponential itself is dimensionless, too. We may also contrary this statement: the declaration of a logarithm should ameliorate move dimensionless together with and so is the logarithm itself.

The key query is: Is the exponential operator \({\mathcal O}\) actually dimensionless?

Of course, the respond is No. It's shocking but the exponential isn't actually dimensionless inward two-dimensional quantum plain theory. Similarly together with to a greater extent than generally, operators inward quantum plain theory almost never take away maintain just the same dimensions that you lot would derive past times the classical dimensional analysis above. How is it possible? And what is the dimension of \({\mathcal O}\)?

Influenza A virus subtype H5N1 funny together with powerful tool inward conformal plain theory – together with our \(D=2\) Klein-Gordon theory is conformal because in that location is no dimensionful parameter anywhere – is the state-operator correspondence. You know that holomorphic functions map the complex bird to a part of some other complex bird together with so that the angles are preserved, right? And the exponential together with logarithm are (locally) holomorphic functions, too. The exponential may map a cylinder to the whole complex bird (without the root at zero).

Note that \(\exp(\sigma^0+ i\sigma^1)\), the exponential of a complex number, is periodic inward \(\sigma^1\) with the periodicity \(2\pi\), OK? This periodicity way that \((\sigma^0,\sigma^1)\) naturally parameterize an infinitely long cylinder. The circle with a constant \(\sigma^0\) that goes unopen to the cylinder gets mapped to a circle unopen to null inward the complex bird past times the exponential, OK? Physics of the Klein-Gordon plain theory on the cylinder together with on the complex bird is basically equivalent.

And a funny thing is that the insertion of an operator at the call for \(z=0\) of the complex bird is equivalent to exciting some terra firma at \(\sigma^0\to -\infty\), at the "infinite past", of the cylinder. Because of the properties of \({\mathcal O} = \exp(ikX)\) nether the translations of \(X\) inward the "target spacetime" \(X\to X+\Delta X\), nosotros may meet that this operator carries some "spacetime momentum", together with the corresponding terra firma has to send the same momentum.

It way that the operator \({\mathcal O} = \exp(ikX)\) is equivalent to a terra firma with the spacetime momentum \(k\). But if the periodicity of \(\sigma^1\) is \(2\pi\), the momentum is the same equally velocity (sorry if in that location is a numerical prefactor) together with the terra firma on the cylinder that nosotros created at \(\sigma^0\to -\infty\) is "moving" equally a business office of the Minkowski globe canvas fourth dimension \(\sigma^0_M\):\[

X(\sigma^0) = X(0) + k \sigma^0_M.

\] It's this simple. We created some terra firma that describes a string at some (delocalized) position but the position is moving with the velocity proportional to the momentum. The fourth dimension nosotros used is the Minkowski globe canvas fourth dimension \(\sigma_M^0\). But inward the truthful conformal plain theory, nosotros demand to convert it to the Euclideanized fourth dimension which way nosotros demand to add together some \(-i\) (to move certain virtually the right sign takes quite some effort, permit me non kicking the bucket there):\[

X(\sigma^0) = X(0) - ik \sigma^0_E.

\] That's great. Let's insert ii copies of the operator \({\mathcal O}\) at ii places of the complex bird – 1 at the root together with some other 1 elsewhere. In the cylinder variables, this corresponds to starting with a momentum \(k\) terra firma inward the infinite past times together with probing it past times some other operator\[

\exp[-ikX(\sigma^0_E)]

\] at some fourth dimension \(\sigma^0_E\). I had to assign the opposite momenta to them (i.e. usage ii operators that are Hermitian conjugates of each other) for the full momentum to vanish, otherwise the expectation value would move zero. But inward the previous displayed equation, nosotros saw the dependence of \(X\) on \(\sigma^0_E\): it was linear with the uncomplicated coefficient (velocity) \(-ik\). So the insertion of the extra operator volition scale like\[

\exp(ik\cdot ik \sigma^0_E) = \exp(-k^2 \sigma^0_E)

\] This is how the expectation value\[

\langle {\mathcal O}(\sigma^0_E=-\infty) {\mathcal O}^\dagger(\sigma^0_E)\rangle

\] depends on \(\sigma^0\). The dependence is the exponential of \(\sigma^0_E\) multiplied past times a constant, namely \(k^2\). But \(\sigma_E^0\) is just the logarithm of the radial coordinate inward the plane,\[

\exp(-k^2 \sigma^0_E) = \exp(-k^2 \ln r) = \frac{1}{r^{k^2}}

\] together with this exponential of a multiple of logarithm may move rewritten equally a uncomplicated mightiness of \(r\). The exponent is \(-k^2\) or \(k^2\) if nosotros write the mightiness inward the denominator. And because this mightiness of \(r\) has to move divided betwixt ii copies of the operator \({\mathcal O}\), nosotros meet that the dimension of \({\mathcal O}\) is \(k^2/2\) – I actually hateful the units are those of \(M^{k^2/2}\). Depending on the coefficient \(k\) inward the exponential, the "spacetime momentum", the exponential plain operator may take away maintain whatever existent non-negative dimension!

Recall that classically, the dimension of the exponential operator \({\mathcal O}\) is zero. Equivalently, when nosotros insert ii operators close to 1 another, in that location is no singularity, cipher special happening when the loci of the ii operators nearly coincide, no diverging prefactor that would move a mightiness of \(r\); the expectation value would all the same scale similar \(1\) equally a business office of the pocket-sized distance betwixt the ii points \(r\). Quantum mechanics adds \(\alpha' k^2/2\) to the (left-moving) dimension [I restored the \(\alpha'\) with units of surface area to allow for SI units inward the spacetime] – together with the corresponding mightiness of \(r\), a novel short-distance singularity, to the two-point function. Note that this nonzero dimension was forced upon us past times quantum mechanics because ii nearby plain operators naturally take away maintain a singularity that scales similar a mightiness police of \(r\) – it's 1 of the manifestations of the short-distance i.e. ultraviolet divergences (which move into subtle to house operators likewise unopen to each other). However, these ultraviolet effects are rather telephone commutation together with straight follow from quantum mechanics because, equally you lot could take away maintain seen, the nonzero exponent straight followed from the mightiness of the exponential operator to create "something that moves inward the target spacetime".

We could take away maintain also deduced the anomalous dimension from the normal ordering of the exponential operator which is needed if you lot kicking the bucket beyond heuristic derivations – but I omitted the normal ordering above. So the nonzero dimension of \(:\exp(ik\hat X):\) doesn't come upwardly from the hats, it actually comes from the colons. ;-) The same normal ordering that makes the normalization of the operator finite (the correlation functions with other operators are finite, non including infinite factors from short-distance divergences) also forces the anomalous dimension on the operator. These ii are inseparable.

You couldn't calculate anything inward perturbative string theory if you lot were ignorant of the fact that the exponential operator has a nonzero dimension. Because the dimension is \(k^2/2\), in that location are rattling interesting novel effects that plow over when \(k^2/2\) is an integer. You tin give the sack take away maintain novel states associated with the exponential operator that bear almost indistinguishably from operators that are just polynomial inward \(X\) together with the globe canvas derivatives. In fact, in that location tin give the sack move \(SU(2)\) symmetries mixing such operators, together with and so on.

Anomalous dimensions together with logarithmic divergences

Let me call a related brain instance whose lesson is a flake different. It's an instance that is to a greater extent than probable to appear inward the "outdated" textbooks of quantum plain theory that are focused on perturbative quantum plain theory inward \(D=4\). Return to \(D=4\) together with consider some quantum plain theory, e.g. the Standard Model. Operators inward such theories also take away maintain anomalous dimensions – quantum corrections to the dimensions you lot would naively calculate past times the classical dimensional analysis. The dimensions of operators may all the same move determined past times looking at the correlation function\[

\langle {\mathcal O}(x,y,z,t) {\mathcal O}^\dagger(x',y',z',t')\rangle

\] which volition take away maintain some extra component of \(1/r^{2\gamma}\), some mightiness of the distance betwixt the ii spacetime points, which volition modify the dimension \(\Delta\) of the operator \({\mathcal O}\) past times the extra term \(\gamma\). How practise these to a greater extent than oft than non fractional powers \(1/r^{2\gamma}\) of the distance betwixt the ii operators emerge inward perturbative quantum plain theory?

The respond is interesting. It's all virtually the logarithmic divergences. Why? Well,\[

\eq{
\langle {\mathcal O}(x,y,z,t) {\mathcal O}(x',y',z',t')\rangle &= \frac{1}{r^{2\gamma}} G\\
&=G (1 -2\gamma \ln r+\dots )
}

\] The funny thing is that if nosotros assume that the anomalous dimension \(\gamma\) is small, together with it should ameliorate move pocket-sized if the plain theory is weakly coupled together with thus good approximated past times the classical plain theory (and its dimensional analysis), nosotros may expand the mightiness of \(r\), \(1/r^{2\gamma} = \exp(-2\gamma \ln r)\), using the Taylor serial together with only alternative the leading together with the initiative of all subleading term. And because the logarithm of the base of operations appears inward the full general mightiness if it is written using an exponential, nosotros larn a term that goes similar \(\ln r\).

Well, equally I mentioned before, arguments of logarithms should actually move dimensionless together with so it's non quite \(r\) that appears in that location if you're careful. The declaration is some production \(r\Lambda\) where \(\Lambda\) has the compensating units of mass. It is some cutoff scale or some other volume scale associated with the physical care for to subtract the short-distance divergences. At whatever rate, you lot meet that the coefficient of \(\ln r\), together with thus a coefficient of \(\ln \Lambda\), is proportional to \(\gamma\), the anomalous dimension.

So if you lot write the correlation business office – including the logarithmically divergent, \(\ln \Lambda\) quantum corrections – inward the format that looks similar the Ansatz above, the coefficient of the logarithmically divergent correction divided past times the leading term is equal to \(-2\gamma\). You may straight extract the anomalous dimension from the logaritmically divergent term (if you lot omit the \(\ln \Lambda\) component itself)!

This is an of import lesson because the logarithmically divergent Feynman diagrams (loop diagrams) tell you lot how the "real dimensions" (dependence of the magnitude on chosen units of length) differ from the naive classical ones. These logarithmically divergent diagrams play the same role equally \(k^2/2\) inward my initiative of all instance from the globe sheet. More generally, logarithmically divergent Feynman diagrams tell you lot "how coupling constants run".

Classically, coupling constants are dimensionless inward \(D=4\). However, quantum mechanically, they're non quite dimensionless – they also take away maintain the anomalous dimension which is opposite to the anomalous dimension of the operators that these coupling constants multiply inward the Lagrangian. We say that "they run". And they only run slowly, logarithmically, which way \(1/g(E_1)^2 - 1/g(E_2)^2 \sim \beta \ln(E_1/E_2)\), together with the speed may move determined from the logarithmically divergent diagrams. The initiative of all ones already start at the one-loop level, together with so they're suppressed past times \(g^2\), ii gauge vertices, relatively to the finite tree-level terms. It way that the scaling of the anomalous dimension is\[

\gamma(e) \sim e^2

\] where \(e\) is a gauge coupling. I chose the missive of the alphabet \(e\) for the electromagnetic coupling, the electron's accuse inward natural units, equally an example. You may roughly say that the operators inward QED take away maintain dimensions that differ from the naive classical ones past times a multiple of \(1/137.036\) because the latter is proportional to \(e^2\), the fine-structure constant! QED is rather weakly coupled but the classical analysis is wrong inward many respects. In particular, fifty-fifty the dimensions of the plain operators may differ roughly past times 1% from your classical estimates.

Again, nosotros meet that the logarithmically divergent Feynman diagrams take away maintain physical consequences – they modify dimensions of operators together with they brand coupling constants run. On the other hand, the mightiness police divergences – fifty-fifty though they facial expression to a greater extent than dramatic because they kicking the bucket to infinity to a greater extent than chop-chop – are less of import together with there's some feel inward which all the mightiness police divergent damage may move just forgotten or laid to zero. The dimensional regularization makes this call for rather self-evident. But the logarithmic divergences, spell seemingly "milder", are "more real". They comprise existent nonzero terms, genuine physical effects, together with every one-loop calculation that completely erases the log divergent damage is leap to brand physically wrong predictions.

Spin plain together with the anomalous dimension \(1/16\)

In the weblog send service virtually the non-Lagrangian plain theories, I also mentioned that the Ising model has the spin plain \(\sigma\) which isn't a polynomial business office of the basic Majorana fermion plain \(\psi\) together with its derivatives. It's clear why it's non one: its dimension is \(\Delta=1/16\). Again, this foreign fractional dimension – the units are volume to the \(1/16\) mightiness – exclusively follows from quantum mechanics.

There are many ways to calculate this dimension but equally I mentioned, a uncomplicated 1 is\[

\eq{
2\Delta &= \zav{ \frac 12 + \frac 32 + \frac 52 + \dots }-(1+2+3+\dots) =\\
&= \frac{1}{24} + \frac{1}{12} = \frac{1}{8}.
}

\] The centre of positive integers is \(-1/12\), the centre of positive half-integers is \(+1/24\), together with the difference betwixt them is \(1/8\) which must move twice the dimension of the spin field. Why? It's because the spin plain is the operator inward the state-operator correspondence that has to move included if you lot desire to switch from the sector of states where \(\psi\) is periodic to the sector where it's antiperiodic or vice versa. So the spin plain \(\sigma\) must take away maintain the activity of mapping the Earth terra firma of the periodic sector to the Earth terra firma of the antiperiodic sector or vice versa. The scaling dimensions of the operators are linked to the energies of states on the cylinder. So the dimension of \(\sigma\) is linked to the break energy difference \(E_{0,P}-E_{0,A}\) betwixt the energies of Earth states inward the periodic together with antiperiodic sectors, respectively. The Earth terra firma energies come upwardly from the zero-point quantum fluctuations together with they're ultimately proportional to the centre of positive integers or half-integers, depending on the sector of the string.

If you lot actually can't swallow that the centre of integers is \(-1/12\) etc. (the "infinite part" nosotros omit when nosotros brand this declaration is a specific instance of the "power police divergences" that may move to a greater extent than oft than non ignored, equally I mentioned above), I assure you lot that in that location are lots of calculations that avoid the centre of integers, formulate it inward a to a greater extent than careful way, or obfuscate its beingness \(-1/12\) inward diverse ways that are non likewise insightful. But if you lot actually know what you're doing together with you lot experience it's to a greater extent than than just some mathematical masturbation, you lot volition concord that it's the finite invariant part of the centre of integers that matters hither together with it's just \(-1/12\).

To summarize, quantum mechanics changes lots of things together with invalidates lots of assumptions that were taken for granted inward classical physics. One of them is the dimensional analysis that was piece of cake to made together with unremarkably implied that all quantities nosotros consider take away maintain units of \(M^\Delta\) where \(\Delta\) is either integer or half-integer. In quantum mechanics, \(\Delta\) may move a fraction such equally \(1/16\) or it may grow with the spacetime momentum of the exponential operators of strings, \(k^2/2\), or it may move proportional to logarithmically divergent diagrams inward quantum plain theory that scale similar \(e^2\) for some coupling constant \(e\).

All these things are belike unavoidably counter-intuitive for a beginner. But they're omnipresent facts together with basic noesis for whatever professional person theoretical (or phenomenological) particle physicist. Those are expected to "start to larn it" some twelvemonth or ii earlier they consummate their PhD – if some "genius who proposes a theory of everything" doesn't larn anomalous dimensions, he's sure less promising a "genius" than a skillful grad student.

In particular, logarithmically divergent diagrams may move the most of import divergent diagrams because they can't move thrown away. Instead, they tell us virtually a novel together with unavoidable, quantum source of the physical phenomena's dependence on the distance scale, quantum corrections to the operators' dimensions etc. These short-distance divergences aren't pathologies. They're rattling interesting physics inward consistent quantum plain theories together with their existence ultimately follows from cipher else than the ordinary nonzero commutators betwixt operators – something that is needed to guarantee that things displace together with quantities evolve with fourth dimension (because \([H,X]\neq 0\)).

The nonzero commutators – e.g. with the Hamiltonian – are needed inward quantum mechanics for quantities to modify with fourth dimension (because of the Heisenberg equations of motion). By the conformal map betwixt the bird unopen to the root together with the cylinder, the dependence on fourth dimension is basically the same equally the dependence on the radial distance – i.e. the distance betwixt ii operators. This dependence is also linked to the commutators. Those are to a greater extent than oft than non nonzero together with interesting. And this dependence on the distance betwixt ii operators inward a production modifies the units of the whole product. That's a way to meet that the anomalous dimensions unavoidably follow from the defining novelty of quantum mechanics, the nonzero commutators. You should actually convey that these phenomena are important, inseparable, together with unavoidable.