Where does the 'i' come from in QFT path integral?

[nuclearhead]

So I've been thinking about the axioms of quantum field theory. In particular the expression for the particle amplitudes:

G(x₁,x₂,...,x_n) = ∫Φ(x₁)Φ(x₂)...Φ(x_n)e^{i S[Φ]/ħ} D[Φ] / ∫Φ(x_n)e^{i S[Φ]/ħ} D[Φ]

But I've been struggling to explain the existence of the 'i'. It seems like this is a postulate that can't be derived from anything else. It makes sense that the exponential is there since we are multiplying an infinite set of amplitudes.

Also, in classical physics the action is only defined up to a multiplicative factor - the Euler-Lagrange equations don't care about a multiplicative factor.

So if we replaced S[Φ] with 100 S[Φ] or 10000 S[Φ] or i S[Φ] we should get the same theory classically. But we should get different quantum field theories right? So how do we determine that there should be this 'i' here? It is often described as a 'phase' but this is not where phase enters into quantum mechanics. Phase comes from, for example, an electromagnetic waves emitted from oscillating source. So this is not really to do with phase.

Is there some kind of experiment which determines that this factor must be i/ħ or can it indeed be something else such as 1000? Or is it a factor like the electron mass or charge that actually does not have a fixed value but depends on your cut off?

One argument is that it comes from Unitarity and compares it to the unitary operator exp(iH) but if the action is, for example S[Φ]=∂Φ∂Φ, which doesn't come from a Hamiltonian then the argument is not valid. As far as I can tell Unitartity is only relevant to fermion wave functions (like Shrodingers equation) because it preserves particles number (and so probability).

For example if you replaced this 'i' with -1, what theoretical result would not then agree with experiment?

It seems if you replace the 'i' with a number α, then you add an extra factor of 1/α to all propagators and α to every interaction. How can this 'i' be derived from experiment?

 

[Avodyne]

That the correct factor is $i/\hbar$ is determined from experiment.

One argument is that it comes from Unitarity and compares it to the unitary operator exp(iH)

Yes, you will not get unitary time evolution without the $i$.

but if the action is, for example S[Φ]=∂Φ∂Φ, which doesn't come from a Hamiltonian

This is wrong. That is the action (actually, the lagrangian density) for a free massless field, and there is a corresponding hamiltonian that you can look up in any QFT book.

 

[atyy]

For example if you replaced this 'i' with -1, what theoretical result would not then agree with experiment?

In fact most calculations do use imaginary time, and so in effect do replace the "i" with -1, so that quantum mechanics becomes statistical mechanics or Euclidean field theory. The i comes about from the usual hand-wavy derivation of the path integral. The path integral for bosons is a sum over classical paths, but quantum mechanics is about operators, Hamiltonians and unitary evolution. Under what circumstances is statistical mechanics secretly about quantum mechanics? In QFT, one answer is given by the Osterwalder-Schrader conditions, the most famous of which is reflection positivity which is needed for unitarity.

http://www.einstein-online.info/spotlights/path_integrals
http://arxiv.org/abs/hep-th/9403084
http://arxiv.org/abs/1005.3751
http://www.itp.uni-hannover.de/saalburg/Lectures/wiese.pdf

 

[nuclearhead]

I'm still not entirely convinced by the arguments. Yes, you need the i for exp(iH) to get a unitary operator. But exp(iS) is not an operator. Its often called a "phase factor".

If S was just an action for free fields, the 'i' doesn't add anything, just an overall phase to the amplitudes which doesn't alter the probabilities. All the probabilities are still conserved so it is "Unitary" in that sense.

It only seems to matter with interaction terms. But at that point, it becomes QFT and can't be derived directly from Hamiltonian quantum mechanics.

I would like to see a specific calculation using Feynman diagrams that shows that you get the wrong answer if it is anything except 'i'. What is the simplest way to show that it leads to something wrong?

For example, can it be shown that if you don't have the 'i' there then electrons would be attracted to electrons instead of repelled?

 

[bhobba]

Check out:
http://arxiv.org/abs/1204.0653

Basically the reason is nearby paths do not cancel without using complex numbers so you do not get the principle of least action - instead you get what is called a Wiener process:
http://en.wikipedia.org/wiki/Wiener_process

Its a very interesting mathematical curiosity that may be telling us something quite important that QM is basically a Wiener process in imaginary time. It certainly its very useful mathematically as shown by a Wick rotation.

Thanks
Bill

 

[nuclearhead]

Hmm... I'll check that out. I'll get to the bottom of this!

It makes me uneasy that there is this complex number at the heart of QFT. For example, most complex numbers in quantum mechanics are not essential, such as we could write a complex spinor as a spinor with twice the components and so on. But there is this one little 'i' that won't go away! We could write it in a two component formalism as [cos(S/ħ),sin(S/ħ)] but this isn't any more satisfying. Or we could do a Wick rotation but then that just transfers the 'i' to the time variable.

It's strange that although the action, which tells you everything you need to know, is real valued, yet to get the probabilities you have to go through this process of introducing 'i'.

It makes you think if an 'i' why not a 'j' and a 'k'? Why not have a quaternion valued QFT which would contain 3 separate actions? e.g. exp(iS₁+jS₂+kS₃) if such a thing even makes sense.

 

[Demystifier]

Yes, you need the i for exp(iH) to get a unitary operator. But exp(iS) is not an operator.

The path integral is derived from the unitary Hamiltonian evolution. As you can see from the derivation
http://en.wikipedia.org/wiki/Relati...ath_integral_formulation_of_quantum_mechanics
"i" in "iS" originates from "i" in "iH".

I would like to see a specific calculation using Feynman diagrams that shows that you get the wrong answer if it is anything except 'i'. What is the simplest way to show that it leads to something wrong?

From Feynman diagrams you can calculate the S-matrix, which must be unitary. Without "i", the S-matrix would not be unitary. In particular, unitarity implies the optical theorem
http://en.wikipedia.org/wiki/Optical_theorem
and without "i" the optical theorem would not be satisfied.

 

[dextercioby]

[...]
From Feynman diagrams you can calculate the S-matrix, which must be unitary. [...]

The normal wisdom is reversed: From the S matrix you calculate (via a perturbative expansion) the Feynman diagrams.

 

[nuclearhead]

My argument is like this. Consider the two point propagator G(x,y) = ∫Φ(x)Φ(y) exp(i∂Φ∂Φ) DΦ
Now absorb the i's into the fields making a substitution Φ' = √(i).

G(x,y) = α ∫Φ'(x)Φ'(y) exp(∂Φ'∂Φ') DΦ'

Surely this gives the same G(x,y) apart from a phase α. (But maybe that phase is undetermined from DΦ' = (i)^∞DΦ)? Perhaps this substitution is forbidden for path integrals?

So by replacing i with -1, for example the propagators get a phase but are still proportional to 1/(x^2-t^2). Assuming an oscillating source at x=0 with energy (equiv. wavelength) E:

∫exp(iEt)/(x^2-(t-t')^2)dt' = exp(iE(t-r))/r θ(E) + exp(iE(t+r))/r θ(-E)

So this is where the phase of waves comes from (nothing to do with the i in the path integral). All the equations of optics are still valid. By replacing i with -1 we still get the same equations for free waves and when we calculate the probabilities the extra phase disappears.

So, like I said the only difference is when replacing i in exp(iS) with -1, say, occurs when we have interacting terms. For an action containing only free fields the 'i' can be absorbed into an overall phase for the propagators. And so everything is still "Unitary" for free fields.

Thus I need convincing that when we have interacting terms the 'i' becomes essential.

As for the Hamiltonian argument, consider that the Shrodinger equation has an 'i' i∂/∂t Ψ = ∇Ψ has an 'i' yet the relativistic Klein-Gordon equation does not. Most derivations of the path integral over fields tend to rely on the classical approximation hence why are we to assume the relativistic S involves the 'i'? If you have a link to deriving the exp(iS) from a relativistic Lagrangian I'd like to see it. Also does not the S-matrix depend explicitly on time? So we would expect the 'i' but S is not dependent on time as it is an integral over all time.

Or the reverse. Is it possible to derive the formula ψ(t₂)=exp(i∫H(t))Ψ(t) from exp(iS)? i.e. how do we recover H from S? That would be a convincing argument.

From Wikipedia (that reliable source!) "The price of a path integral representation is that the unitarity of a theory is no longer self-evident, but it can be proven by changing variables to some canonical representation." Do you know where I can read about this? It seems to be the answer.

 

[nuclearhead]

OK... let me see if I've got this right:

For the Klein-Gordon field the Hamiltonian is:

H = ∫dx ³ (π(x)² + ∇²Φ + m²Φ)

So to have a Unitary operator we must have U = exp(iH)

And then going from the Hamiltonian to the action we do a Fourier transform exp(iS) = ∫ exp(i∫H(t)dt) exp( i π ∂/∂t Φ) D[π]

Hmm... I guess that 'i' really is there to stay!

And I guess to get the Hamiltonian from the action is something like:

H[Φ(t),π(t)] =π δS/δ(∂_tΦ) ??

 

[nuclearhead]

OK. But now I have a new confusion.

If the Klein Gordon equation is a real field. How can you have a Unitary operator on a real field?

 

[Jazzdude]

OK. But now I have a new confusion.

If the Klein Gordon equation is a real field. How can you have a Unitary operator on a real field?

The classical KG field equation is for a real field. After field quantisation you get a field of operators in the Heisenberg picture. Those evolve unitarily.

Cheers,

Jazz

 

[nuclearhead]

I still need to delve deeper. These are the things that keep me up at night! If I don't fully understand things. But perhaps no-one really does.

Lets say we have defined our theory only in terms of waves and positions. Then we find of our wave functions have wavelike behaviour. So we define a wavelength or energy as the Fourier transform of the wave-functions. Thereby introducing 'i'. We find we can replace d/dt in our equations with iE. Now we postulate that energy 'E' is conserved over time. Hence all our fundamental theory is now based on iE.

So it seems that we have this 'i' because we are thinking in terms of a Fourier transform of the wave functions. It strikes me that there must be an equivalent way of writing the equations that avoids this Fourier transform in the first place? Perhaps a form that avoids complex numbers?

Didn't we just introduce i to nicely represent sinusoidal waves of light? Yet, the conservation of momentum, etc. is fundamentlly in the Fourier transformed picture. This is hurting my head.

In the position picture does momentum conservation become Lorenz invariance? And so energy conservation becomes...?

 

[nuclearhead]

I've had a Damascus moment! What I realized is that the 'i' is not an accidental hangover from writing spinors as complex numbers. But the reverse! Spinors are accidentally able to be written in complex numbers due to them being linear in the derivatives. Therefore I now realize the complex (or equiv. two component) formalism of QFT is the very essence of the theory! The fact that the Dirac or Shrodinger equation for electrons can be written in complex form is an accidental consequence of this complex structure of the path integral. So now that I've accepted the complex structure I am now ready to get rid of it! What I mean is that I only want to think of QFT without ever using the term "complex amplitude". Thus if I wanted to explain QFT to an alien who wasn't prepared to accept the concept of √(-1) I would explain it like this:

The "probability density" to find particles at x₁,x₂,... on the boundary of a causal surface is:

P(x₁,x₂...) = (∫F[Φ]cos(S[Φ])D[Φ])² + (∫F[Φ]sin(S[Φ])D[Φ])²

where F[Φ]=Φ(x₁)Φ(x₂)...​

This makes me somewhat happy. Since it is an equation which avoids the use of complex numbers. True it is not as compact and it says exactly the same thing. Plus it has a nice kind of Pythagorean quality to it.

You might think, what kind of Alien would not know about 'i' but be able to understand functional integrals.

Anyway, it is a fun game trying to rewrite QFT without complex numbers. Another one I came up with is the propagator:

1/(k.k+iε) can also be replaced with k.k/( (k.k)²+ε²) to avoid the iε

I'm not quite sure how you would rewrite the commutation rules [x,p]=iħ or the equation i∂/∂tΦ=HΦ or if you even need these. It can probably be done though.

 

[vanhees71]

You should derive the path integral from the operator formulation for one simple model (e.g., simple $\phi^4$ theory) to see, where the factors come from. Further note that quantum field theory means that the fields get operators. The most simple way to achieve this is to use what's called "canonical quantization". For an introduction see

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

There you also find the path-integral formalism for non-relativsitic "first-quantized" quantum mechanics. I think it also helps to first consider this more simple case, before tackling the pretty involved QFT formalism.

 

[nuclearhead]

Yes, the canonical formalism helped me see how the iS is needed for unitarity. The Klein Gordon equation can be thought of as independent fields Φ and ∂₀Φ for the purpose of quantisation.

It's a pity that exp(iS) doesn't converge properly without doing some magic like a Wick rotation. Are there other ways to make it converge? Such as adding a small factor exp(iS-εS²) or integrating the fields over a finite size Universe?

 

[atyy]

Yes, the canonical formalism helped me see how the iS is needed for unitarity. The Klein Gordon equation can be thought of as independent fields Φ and ∂₀Φ for the purpose of quantisation.

It's a pity that exp(iS) doesn't converge properly without doing some magic like a Wick rotation. Are there other ways to make it converge? Such as adding a small factor exp(iS-εS²) or integrating the fields over a finite size Universe?

But doesn't the Wick rotation answer your original question affirmatively - the i in the path integral is in some sense not needed. We can recover the i in the canonical formalism without the i in the path integral.

 

[vanhees71]

In principle you are right, but Euclidean field theory is not the full story. In principle you can analytically continue back to real times, leading you to the retarded Green's functions, and this is in principle sufficient to reconstruct also the time-ordered Green's functions in the vacuum (and the full Keldysh Green's functions in thermal equilibrium). However, this is not an easy task in practice. E.g., in QCD the rigorous way to gain information is through lattice-gauge calculations. For the case of finite-temperature QCD that's a Euclidean path integral with the Matsubara boundary conditions in the imaginary-time coordinate. This in principles gives the Matsubara correlation functions, but it's very tough to analytically continue those to the corresponding real-time quantities, which are sometimes of high interest in heavy-ion physics. E.g., any transport coefficients in the Kubo formalism need the time-like limit of a two-point function in momentum space, i.e., you need a reliable analytical continuation of the Matsubara two-point function to a real-time retarded Green's function in the neighborhood of a quite singular point in the complex energy plane.

For the usual purposes in the vacuum case and to derive Feynman rules, all you need is an infinitesimal Wick rotation, not the full 90 degrees. This is closely related to the "adiabatic-switching procedure" a la Gell-Mann and Low in the operator formalism. It's very important to understand adiabatic switching and the meaning of the infinitesimal Wick rotation in the real-time path-integral formalism. The trouble is, that all this is (to my knowledge) not really rigorously provable from a mathematical point of view. Usually physicists have quite "robust" hand-waving treatments of all these subtle issues. The, in my opinion, best explanation of this (and the LSZ-reduction formalisms) is in Bailin and Love, Gauge Theories (for the path-integral approach) and in Srednicky (for the operator formalism).

 

[nuclearhead]

Well, true you could wick rotate but then you would get propagators in G(x,y,z,it).

It would be nicer to have a limit in which you stuck with Minkowski space-time. Well, I think it would be nicer anyway. :)

I'm thinking about if you formalized all of QFT in a computer, the computer might have trouble with all the Wick rotating or summing of sine waves that don't converge. I'm sure there is a nice mathematically precise way of defining the limit. Although for practical purposes the Wick rotation would be quicker.

 

[atyy]

It would be nicer to have a limit in which you stuck with Minkowski space-time.

I'm thinking about if you formalized all of QFT in a computer, the computer might have trouble with all the Wick rotating or summing of sine waves that don't converge. I'm sure there is a nice mathematically precise way of defining the limit.

The Osterwalder-Schrader axioms are one way of making sure that the Euclidean field theory is equivalent to QFT in Minkowski spacetime. They have been used to construct rigourous relativistic field theories in 2 and 3 spacetime dimensions. The Osterwalder-Schrader axioms are listed in http://www.rivasseau.com/resources/book.pdf (p17-18).

The rigourous construction of a relativistic field theory in 4 spacetime dimensions is an open problem. For example, no one has constructed a rigourous QCD. Also, as vanhees71 points out above, even if it were possible to do so in principle by Euclidean field theory, the imaginary time calculation is not necessarily easier in practice, especially for non-equilibrium processes.

 

[nuclearhead]

I'm glad I'm not the only one who has a problem with the current formalism then!

 

[atyy]

I'm glad I'm not the only one who has a problem with the current formalism then!

There is no non-perturbative definition of the standard model at the moment, in other words, one has perturbative "approximations", except they aren't real approximations since we don't know what they are approximating.

There are two methods to try to get a non-perturbative formulation. The first is constructive field theory using Euclidean field theory, which gets exactly relativistic field theory. However, there is no known 4D construction.

The second method is to give up exact relativity, and put the theory on a lattice in a large but finite volume so that although the theory is not relativistic, it will be well-defined. However, whether one can put the standard model chiral fermions interacting with non-abelian gauge fields on the lattice is unsolved.

If one has a lattice model from the second method, and if one is able to take the lattice spacing to zero and extend the lattice to an infinite volume, then the second method can be the basis of a relativistic field theory constructed using the first method.

 

[bhobba]

It's a pity that exp(iS) doesn't converge properly without doing some magic like a Wick rotation.

A wick rotation has got nothing to do with convergence (a simple change of variables can't affect convergence) which in the path integral approach is a difficult issue. It is only of recent times it has been solved by means of so called Hida distributions from white noise analysis:
http://arxiv.org/pdf/0805.3253v1.pdf.
http://mathlab.math.scu.edu.tw/mp/pdf/S20N41.pdf

The math is quite advanced, but that's because the issue is difficult - best not to worry about it at the start.

Thanks
Bill

 

[bhobba]

the i in the path integral is in some sense not needed.

Its critical - its needed so path cancellation occurs and you get the PLA instead of a Wiener process.

Thanks
Bill

 

[atyy]

A wick rotation has got nothing to do with convergence (a simple change of variables can't affect convergence) which in the path integral approach is a difficult issue. It is only of recent times it has been solved by means of so called Hida distributions from white noise analysis:
http://arxiv.org/pdf/0805.3253v1.pdf.
http://mathlab.math.scu.edu.tw/mp/pdf/S20N41.pdf

Its critical - its needed so path cancellation occurs and you get the PLA instead of a Wiener process.

The Wick rotation does affect the convergence, but it also changes it to something more like a Wiener process. But surprisingly, under some circumstances a Wiener process or statistical mechanics or Euclidean field theory is also exactly a quantum field theory in Minkowski spacetime.

 

[dx]

Consider throwing a coin. If there are N possible microstates, then by the symmetry of the coin, half of these will land H and half of these will land T. The fact that the probability of H or T is half is based on the following two assumptions:

1. The coin throw assigns equal probability to each microstate, i.e. P = exp ( - log N )
2. Half the microstates will land H

If we have two microstates A and B which lead to an outcome C, then

P(C) = P(A) + P(B)

This rule is based on the assumption that if C happens, then either A or B must have happened. This assumption does not hold in quantum theory. It is replaced by the rule

φ(C) = φ(A) + φ(B)

where the φ are complex numbers called amplitudes. The probabilities are the squares of these complex numbers. The complex number feynman amplitude exp(iS) is the quantum analog of the real number probability exp (- log N) which we have introduced above in the context of classical probability.

P(C) = ∫ exp ( - log N) [schematic]

becomes

φ(C) = ∫ exp ( iS ) [schematic]

 

[dx]

Also, an interesting observation from the last two formulas in my post is that the action S is sort of analogous to the entropy log N (also usually called S!)

This is important and deep, but not fully understood. It is related to the idea that temperature is related to the imaginary periodicity of time.

 

[vanhees71]

For the relation between finite-temperature QFT and the imaginary-time (Matsubara) formalism for equilibrium QFT (in the grand canonical ensemble), see my lecture notes on (relativistic) many-body theory. The relation is of course the same for the nonrelativistic case:

http://fias.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

 

[stevendaryl]

A wick rotation has got nothing to do with convergence (a simple change of variables can't affect convergence)

It's not simply a change of variables. Look at a simple example:

$F(t) = \sum_j e^{-i E_j t/\hbar}$

where the sum is over all states of some system, and $E_j$ is the energy of state $j$. In general, this sum doesn't converge. But if we make the substittution

$\beta = it/\hbar$

then we get the sum

$G(\beta)= \sum_j e^{-\beta E_j}$

This is perfectly well defined (if $E_j \rightarrow \infty$ as $j \rightarrow \infty$) for all real values of $\beta$. Then if we can evaluate $G(\beta)$, then we can (maybe) analytically continue it to the case $\beta = -i \hbar t$ to find $F(t)$.

So, it's not just a change of variables, it's a change of variables plus an analytical continuation.

 

[bhobba]

So, it's not just a change of variables, it's a change of variables plus an analytical continuation.

The fact you can do that change of variables implies analytic continuation anyway. Its not legitimate to substitute a complex value into an equation of real variables although its often done - the assumption being you have, via analytic continuation, extended the function to the complex plane.

But that's not what's going on here. In QM it is assumed to be defined on the complex plane anyway. A path integral has convergence issues before or after a wick rotation.

In particular in your example you are assuming t is real then substituting a complex value into it.

Thanks
Bill

 

[stevendaryl]

The fact you can do that change of variables implies analytic continuation anyway. Its not legitimate to substitute a complex value into an equation of real variables although its often done

Suppose that you're in the following situation (well, maybe not you, but somebody else):

• You are trying to compute some physically meaningful quantity $F(t)$
  
• Your theoretical analysis gives you a mathematical expression, $\sum_n e^{-i \frac{E_n t}{\hbar}}$
• This expression does not converge for $t$ real.
• Therefore, it cannot literally be the case that $F(t) = \sum_n e^{-i \frac{E_n t}{\hbar}}$, since the left-hand side is something meaningful, and the right-hand side is nonsense.
  
• However, the similar expression $\sum_n e^{-\beta E_n}$ does converge, when $\beta$ is real and positive, to a function $\tilde{F}(\beta)$.
• The function $\tilde{F}(\beta)$ can be analytically continued to the region where $\beta$ is purely imaginary.

So you hypothesize that the $F(t) = \tilde{F}(\frac{i t}{\hbar})$

I'm not sure which step you're saying is not legitimate. There is no claim that the original sum converges to $\tilde{F}(\frac{it}{\hbar})$. It certainly doesn't. The claim is that the physically meaningfully value $F(t)$ is equal to $\tilde{F}(\frac{it}{\hbar})$. That's a hypothesis, not a conclusion, so the notion of "legitimate" versus "illegitimate" doesn't come up. The only issue is whether that way of computing $F(t)$ agrees with observation.

 

[atyy]

I looked at the Hida distributions approach that bhobba linked to, and they are closer to what he is saying. I think stevendaryl is thinking more of the Wick rotation which changes the problem from the hand-wavy derivation of the real time path integral to a statistical mechanics problem via imaginary time, which is invalid from the point of view of the hand-wavy derivation, yet can be shown to rigourously define relativistic quantum field theory under some conditions.

 

[bhobba]

I'm not sure which step you're saying is not legitimate.

It not legitimate to substitute a complex variable for a real t unless the equation is extended by analytic continuation.

But that is beside the point. Both Wiener processes and path integrals have convergence issues that are solved by Hida Distributions.

Thanks
Bill

 

[stevendaryl]

I looked at the Hida distributions approach that bhobba linked to, and they are closer to what he is saying. I think stevendaryl is thinking more of the Wick rotation which changes the problem from the hand-wavy derivation of the real time path integral to a statistical mechanics problem via imaginary time, which is invalid from the point of view of the hand-wavy derivation, yet can be shown to rigourously define relativistic quantum field theory under some conditions.

Yes, I'm talking about path integrals. They don't literally converge to anything. The way that people get meaningful results out of them is by analytically continuing similar expressions that do converge.

Even for the simplest case of the free one-dimensional, nonrelativistic, massive spin-zero particle, evaluating the path integrals involve expressions such as:

$\int e^{(i k x - i t \frac{\hbar k^2}{2m})} dk$

That expression doesn't converge. However, the related expression

$\int e^{(k u - \beta \frac{\hbar^2 k^2}{2m})} dk$

does converge, to $F(u, \beta) = \frac{2m \pi}{\beta \hbar^2} e^{- \frac{mu}{2\beta\hbar^2}}$ (if I did that right). So the assumption is that the free propagator is actually obtained by replacing $u$ by $ix$ and $\beta$ by $i \frac{t}{\hbar}$ in $F(u,\beta)$.

 

[stevendaryl]

It not legitimate to substitute a complex variable for a real t unless the equation is extended by analytic continuation.

What do you mean by "legitimate"? As I said, there is no mathematical claim being made. It's a physical claim that a particular physically measurable quantity is equal to a particular mathematical expression.