What is the basic scheme of quantum field theories?

[jonjacson]

In classical mechanics I would say:
a particle can have any initial position and velocities, Newton laws give you the evolution of the particle:
F=ma , is the basic equation, if you know the forces acting on the particle by solving this equation you get the future values for velocity and position

In quantum mechanics:
The "particle" is characterized by a wavefunction that can be used to calculate the probability of finding the particle at certain position or momentum.
If you know the wavefunction you plug it into the Schrodinger equation and you get the evolution on time of this wavefunction.

Can anybody tell me a similar basic scheme for quantum field theory? For example, in QED, Is there a basic differential equation? What about QCD?

Thanks for your time.

 

[MathematicalPhysicist]

Dirac equation and Klein-Gordon equation.
They appear in RQM, they also appear in QFT but there we have Ward-Takahashi identity which is a generalization of Noether's theorem.

 

[jonjacson]

Dirac equation and Klein-Gordon equation.
They appear in RQM, they also appear in QFT but there we have Ward-Takahashi identity which is a generalization of Noether's theorem.

So if you know how to solve the dirac equation you can predict the future behavior of the particle?

 

[meopemuk]

QFT does not allow you to calculate the time evolution, because the Hamiltonian (=the generator of time evolution) of QFT is badly damaged by the presence of divergent counterterms. QFT can only calculate the S-matrix, which is a mapping of states from the infinite past to the infinite future. From the calculated S-matrix you can extract scattering cross-sections and energies/lifetimes of bound states. That's all you can do with the textbook QFT.

"The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. From the physical point of view, this is not so surprising, because in contrast to non-relativistic quantum mechanics, the time behavior of a relativistic system with creation and annihilation of particles is unobservable. Essentially only scattering experiments are possible, therefore we retreat to scattering theory. One learns modesty in field theory." G. Scharf, Finite quantum electrodynamics. The causal approach, 1995.

Eugene.

 

[jonjacson]

QFT does not allow you to calculate the time evolution, because the Hamiltonian (=the generator of time evolution) of QFT is badly damaged by the presence of divergent counterterms. QFT can only calculate the S-matrix, which is a mapping of states from the infinite past to the infinite future. From the calculated S-matrix you can extract scattering cross-sections and energies/lifetimes of bound states. That's all you can do with the textbook QFT.

"The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. From the physical point of view, this is not so surprising, because in contrast to non-relativistic quantum mechanics, the time behavior of a relativistic system with creation and annihilation of particles is unobservable. Essentially only scattering experiments are possible, therefore we retreat to scattering theory. One learns modesty in field theory." G. Scharf, Finite quantum electrodynamics. The causal approach, 1995.

Eugene.

Do you mean there is more QFT than what I can find on a textbook?

Your answer is what I was looking for, thank you so much to clarify a little bit what that theory is.

 

[jonjacson]

QFT does not allow you to calculate the time evolution, because the Hamiltonian (=the generator of time evolution) of QFT is badly damaged by the presence of divergent counterterms. QFT can only calculate the S-matrix, which is a mapping of states from the infinite past to the infinite future. From the calculated S-matrix you can extract scattering cross-sections and energies/lifetimes of bound states. That's all you can do with the textbook QFT.

"The more one thinks about this situation, the more one is led to the conclusion that one should not insist on a detailed description of the system in time. From the physical point of view, this is not so surprising, because in contrast to non-relativistic quantum mechanics, the time behavior of a relativistic system with creation and annihilation of particles is unobservable. Essentially only scattering experiments are possible, therefore we retreat to scattering theory. One learns modesty in field theory." G. Scharf, Finite quantum electrodynamics. The causal approach, 1995.

Eugene.

And I am thinking about it, you must be able to reproduce any calculation made by the simpler quantum mechanics theory right? If so, how do you explain the double slit experiment? For example.

In quantum mechanics the wavefunction interferes with itself, is that correct?

What is the explanation of the experiment according to qft?

 

[meopemuk]

Do you mean there is more QFT than what I can find on a textbook?

To go beyond textbooks you can try "E. Stefanovich, Elementary particle theory. Vol. 1-3, De Gruyter, Berlin. 2018".
The idea there is to reformulate QFT in a language similar to ordinary quantum mechanics: with wave functions, finite Hamiltonians, and well-defined time evolution.

Eugene.

 

[jonjacson]

To go beyond textbooks you can try "E. Stefanovich, Elementary particle theory. Vol. 1-3, De Gruyter, Berlin. 2018".
The idea there is to reformulate QFT in a language similar to ordinary quantum mechanics: with wave functions, finite Hamiltonians, and well-defined time evolution.

Eugene.

I thought doing that was not possible according to your previous post.

Anyway I will purchase those three books and have a look.

Your help is very helpful, thanks.

 

[PeterDonis]

Moderator's note: thread level changed to "I".

 

[jonjacson]

Eugene, according to Stefanovich special relativy is violated and fields are just particles? They say Pizella observed that in the laboratory.

 

[PeterDonis]

I thought doing that was not possible according to your previous post.

I think he meant that the standard way of doing QFT does not allow you to do that. Stefanovich's textbook, as I understand it, is an alternate approach to the entire subject.

Can anybody tell me a similar basic scheme for quantum field theory?

Not if by "basic scheme" you mean "describe the system by some function that evolves in time according to a differential equation".

If you think about it, such a scheme can't work as it stands in a relativistic theory, because "evolves in time" presupposes that "time" has a definite, absolute meaning. But in relativity, it doesn't; only spacetime is absolute, space and time are relative. (To put it another way, describing a relativistic system by a function that evolves in time according to a differential equation would mean picking out some particular reference frame as the "real" or "preferred" one, which is inconsistent with the principle of relativity.)

What standard QFT does, instead, is to redescribe everything in terms of operators that act at particular events, i.e., points in spacetime. In other words, "quantum fields" are not like ordinary fields that you're used to: they're not numbers that describe the state of something at a particular time (or time and place). They are operators that act at particular points in spacetime to produce measurement results.

QFT can only calculate the S-matrix

I think this is a bit pessimistic. There is a lot of work done in non-perturbative QFT and studying effects that cannot be modeled by an S-matrix.

 

[PeterDonis]

you must be able to reproduce any calculation made by the simpler quantum mechanics theory right?

Yes, but the way this is done is to show that, under appropriate conditions, standard non-relativistic QM is a valid approximation to QFT. Then you just show that those conditions apply to a particular problem (such as the double slit) and do the calculation using standard non-relativistic QM just as you did before. Nobody tries to calculate things using full-blown QFT that can be adequately calculated using standard non-relativistic QM; it would just take a lot more time and effort to get the same answer.

 

[jonjacson]

Yes, but the way this is done is to show that, under appropriate conditions, standard non-relativistic QM is a valid approximation to QFT. Then you just show that those conditions apply to a particular problem (such as the double slit) and do the calculation using standard non-relativistic QM just as you did before. Nobody tries to calculate things using full-blown QFT that can be adequately calculated using standard non-relativistic QM; it would just take a lot more time and effort to get the same answer.

But for an student it would be very useful to grasp the topic, I guess.

 

[jonjacson]

I think he meant that the standard way of doing QFT does not allow you to do that. Stefanovich's textbook, as I understand it, is an alternate approach to the entire subject.
Not if by "basic scheme" you mean "describe the system by some function that evolves in time according to a differential equation".

If you think about it, such a scheme can't work as it stands in a relativistic theory, because "evolves in time" presupposes that "time" has a definite, absolute meaning. But in relativity, it doesn't; only spacetime is absolute, space and time are relative. (To put it another way, describing a relativistic system by a function that evolves in time according to a differential equation would mean picking out some particular reference frame as the "real" or "preferred" one, which is inconsistent with the principle of relativity.)

What standard QFT does, instead, is to redescribe everything in terms of operators that act at particular events, i.e., points in spacetime. In other words, "quantum fields" are not like ordinary fields that you're used to: they're not numbers that describe the state of something at a particular time (or time and place). They are operators that act at particular points in spacetime to produce measurement results.
I think this is a bit pessimistic. There is a lot of work done in non-perturbative QFT and studying effects that cannot be modeled by an S-matrix.

Well by basic scheme I mean what you say at the end of your post, when you start with: "What standard QFT does, instead...:" , that is what I am interested in. In having a global picture, or approximate picture on the theory.

 

[DarMM]

QFT says that the fundamental quantities you can observe are fields, e.g. a scalar field $\phi(x)$ and also functions of those fields and their derivatives e.g. $\phi^{2}\partial_{\mu}\phi(x)$. Here function can also mean integrals of the fields, included weighted integrals where the field is weighted against a function e.g. $\int{\phi^{3}(x)g(x)d^{4}x}$. Being mathematically precise only such weighted integrals are valid observables.

The theory describes the statistics of such field observables by assigning to each observable, $A$, an expectation value $<A>$. This is accomplished by the state $\rho$, a mathematical machine that maps each observable to its expectation: $\rho: A \rightarrow <A>$. If this $\rho$ is a pure state (minimum entropy), it is equivalent to an element of a vector space called a Hilbert space and then we have $\rho(A) = (\psi,A\psi)$, with $A$ now an operator on this Hilbert space. (I'm assuming you know QM here given the I level)

The most interesting pseudo-expectation values are correlation functions, expectations of the fields at different points, e.g. $\rho(\phi(x)\phi(y))$. I say pseudo-expectation value (not a standard term, just for exposition here) because as above unweighted fields are not strictly observable, but nevertheless you obtain much of the physics from calculating these correlation functions.

Time evolution is described via the Hamiltonian observable which is (like all observables) a function of the fields.

We have a method known as perturbation theory which allows us to compute the correlation functions.

However for most situations in QFT the calculations are prohibitively difficult.

The most tractable situation is one involving very simple states which in the far past or far future look like states from a free field theory. These asymptotically free states are often called particles, though they only behave like particles at asymptotic times. One then has a quantity, the S-matrix. The S-matrix roughly has the form $S_{nm}$, whose modulus squared is the chance to go from $n$-th in-state $m$-th out-state.

Via something called the LSZ formula, it turns out the S-matrix can be obtained from the correlation functions. From above, this ultimately means particle to particle transitions can be computed via perturbation theory.

So we have these states that look like particles in the far past and the far future. We can prepare the former in many ways (e.g. proton beam) and we can detect the latter in many ways (e.g. detectors in the LHC). Thus this particular set up is both mathematically and experimentally tractable and it is how we have tested QFT in the main.

The case of looking at things at finite times is much less developed. Also bound states like Hydrogen. Even for the simple particle-to-particle case above perturbation theory doesn't always work if the interactions are too strong. We know that in general even the concept of particle breaks down in these theories and can be observer dependent, however up to now this has mattered little due to the way we test the theories.

Just like QM, QFT seems to be simply a probability calculus for observing quantities that are somewhat like classical fields in form, or in certain limited situations particles. I say "somewhat" because their statistics are radically different. However no "narrative" is provided as to what (if anything) generates those statistics. That's the old issue of interpreting QM, for which QFT seems to provide little additional insight currently.

 

[A. Neumaier]

To go beyond textbooks you can try "E. Stefanovich, Elementary particle theory. Vol. 1-3, De Gruyter, Berlin. 2018".
The idea there is to reformulate QFT in a language similar to ordinary quantum mechanics: with wave functions, finite Hamiltonians, and well-defined time evolution.

You forgot to say that this is your book, with a nonstandard reformulation that sacrifices Lorentz invariance and allows faster than light phenomena...

 

[jonjacson]

QFT says that the fundamental quantities you can observe are fields, e.g. a scalar field $\phi(x)$ and also functions of those fields and their derivatives e.g. $\phi^{2}\partial_{\mu}\phi(x)$. Here function can also mean integrals of the fields, included weighted integrals where the field is weighted against a function e.g. $\int{\phi^{3}(x)g(x)d^{4}x}$. Being mathematically precise only such weighted integrals are valid observables.

The theory describes the statistics of such field observables by assigning to each observable, $A$, an expectation value $<A>$. This is accomplished by the state $\rho$, a mathematical machine that maps each observable to its expectation: $\rho: A \rightarrow <A>$. If this $\rho$ is a pure state (minimum entropy), it is equivalent to an element of a vector space called a Hilbert space and then we have $\rho(A) = (\psi,A\psi)$, with $A$ now an operator on this Hilbert space. (I'm assuming you know QM here given the I level)

The most interesting pseudo-expectation values are correlation functions, expectations of the fields at different points, e.g. $\rho(\phi(x)\phi(y))$. I say pseudo-expectation value (not a standard term, just for exposition here) because as above unweighted fields are not strictly observable, but nevertheless you obtain much of the physics from calculating these correlation functions.

Time evolution is described via the Hamiltonian observable which is (like all observables) a function of the fields.

We have a method known as perturbation theory which allows us to compute the correlation functions.

However for most situations in QFT the calculations are prohibitively difficult.

The most tractable situation is one involving very simple states which in the far past or far future look like states from a free field theory. These asymptotically free states are often called particles, though they only behave like particles at asymptotic times. One then has a quantity, the S-matrix. The S-matrix roughly has the form $S_{nm}$, whose modulus squared is the chance to go from $n$-th in-state $m$-th out-state.

Via something called the LSZ formula, it turns out the S-matrix can be obtained from the correlation functions. From above, this ultimately means particle to particle transitions can be computed via perturbation theory.

So we have these states that look like particles in the far past and the far future. We can prepare the former in many ways (e.g. proton beam) and we can detect the latter in many ways (e.g. detectors in the LHC). Thus this particular set up is both mathematically and experimentally tractable and it is how we have tested QFT in the main.

The case of looking at things at finite times is much less developed. Also bound states like Hydrogen. Even for the simple particle-to-particle case above perturbation theory doesn't always work if the interactions are too strong. We know that in general even the concept of particle breaks down in these theories and can be observer dependent, however up to now this has mattered little due to the way we test the theories.

Just like QM, QFT seems to be simply a probability calculus for observing quantities that are somewhat like classical fields in form, or in certain limited situations particles. I say "somewhat" because their statistics are radically different. However no "narrative" is provided as to what (if anything) generates those statistics. That's the old issue of interpreting QM, for which QFT seems to provide little additional insight currently.

Great, this is the stuff I wanted to read when I opened the thread. Thank you so much.

In regards of the complexity of solving this equations, I have found a book called "molecular quantum electrodynamics". I guess dramatic aproximations will be made in order to use this theory for N body systems, Is this correct?

So QED is so difficult that it is not used in biochemistry, Am I correct?

 

[jonjacson]

You forgot to say that this is your book, with a nonstandard reformulation that sacrifices Lorentz invariance and allows faster than light phenomena...

This post was totally unexpected, interesting.

 

[A. Neumaier]

This post was totally unexpected, interesting.

The book gives a minority view based on an extreme distrust (and lack of understanding) of field theory.

So QED is so difficult that it is not used in biochemistry

One uses simpler approximations (deduced from QED) in all of quantum chemistry, whether bio or not. Primarily nonrelativistic approximations with some relativistic corrections (where needed). This already gives a Hamiltonian multiparticle description - fields only contribute to the terms in the Hamiltonian. Then further simplification using theBorn-Oppenheimer approximation, then more and more further approximations the bigger the number of electrons.

 

[atyy]

In quantum mechanics:
The "particle" is characterized by a wavefunction that can be used to calculate the probability of finding the particle at certain position or momentum.
If you know the wavefunction you plug it into the Schrodinger equation and you get the evolution on time of this wavefunction.

In condensed matter, quantum field theory is simply an equivalent way of formulating quantum mechanics for many identical particles. The translation between the quantum mechanics you have described and the quantum mechanics of quantum field theory is called "second quantization" (a confusing name, used for historical reasons).
https://phy.ntnu.edu.tw/~changmc/Teach/SM/ch01.pdf
http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-2-qftc3.pdf

 

[A. Neumaier]

In condensed matter, quantum field theory is simply an equivalent way of formulating quantum mechanics for many identical particles.

But this does not apply for QED and QCD, which appear in post #1.

 

[atyy]

But this does not apply for QED and QCD, which appear in post #1.

https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf

 

[jonjacson]

The book gives a minority view based on an extreme distrust (and lack of understanding) of field theory.

One uses simpler approximations (deduced from QED) in all of quantum chemistry, whether bio or not. Primarily nonrelativistic approximations with some relativistic corrections (where needed). This already gives a Hamiltonian multiparticle description - fields only contribute to the terms in the Hamiltonian. Then further simplification using theBorn-Oppenheimer approximation, then more and more further approximations the bigger the number of electrons.

Thank you so much.

In condensed matter, quantum field theory is simply an equivalent way of formulating quantum mechanics for many identical particles. The translation between the quantum mechanics you have described and the quantum mechanics of quantum field theory is called "second quantization" (a confusing name, used for historical reasons).
https://phy.ntnu.edu.tw/~changmc/Teach/SM/ch01.pdf
http://users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-2-qftc3.pdf

Nice reads, thanks!

 

[Demystifier]

In classical mechanics I would say:
a particle can have any initial position and velocities, Newton laws give you the evolution of the particle:
F=ma , is the basic equation, if you know the forces acting on the particle by solving this equation you get the future values for velocity and position

In quantum mechanics:
The "particle" is characterized by a wavefunction that can be used to calculate the probability of finding the particle at certain position or momentum.
If you know the wavefunction you plug it into the Schrodinger equation and you get the evolution on time of this wavefunction.

Can anybody tell me a similar basic scheme for quantum field theory? For example, in QED, Is there a basic differential equation? What about QCD?

In QFT, the wave function gets generalized to a wave functional. If the wave function at a fixed time is $\psi({\bf x})$, then the wave functional at a fixed time is $\Psi[\phi]$, where $\phi$ is the set of values of the field $\phi({\bf x})$ at all points ${\bf x}$. There is a differential equation (the so called functional Schrodinger equation) that describes how $\Psi[\phi]$ changes with time, in very much same the way as the ordinary Schrodinger equation describes how $\psi({\bf x})$ changes with time.

 

[jonjacson]

In QFT, the wave function gets generalized to a wave functional. If the wave function at a fixed time is $\psi({\bf x})$, then the wave functional at a fixed time is $\Psi[\phi]$, where $\phi$ is the set of values of the field $\phi({\bf x})$ at all points ${\bf x}$. There is a differential equation (the so called functional Schrodinger equation) that describes how $\Psi[\phi]$ changes with time, in very much same the way as the ordinary Schrodinger equation describes how $\psi({\bf x})$ changes with time.

WHy is that each one of you gives me a different version?

I will search info about wave functionals.

I appreciate any comment really, thank you so much.

 

[Demystifier]

WHy is that each one of you gives me a different version?

Because there are many equivalent ways to represent QFT. The one I mentioned is the closest to the Schrodinger picture of ordinary QM. But it is not very convenient in practice, so practical physicists usually ignore it.

 

[atyy]

WHy is that each one of you gives me a different version?

I will search info about wave functionals.

https://www.amazon.com/dp/0201360799/?tag=pfamazon01-20 talks about the Schroedinger picture wave functional formulation (mentioned by @Demystifier), and its equivalence to the more usual Heisenberg picture formulation of QFT.

 

[jonjacson]

Because there are many equivalent ways to represent QFT. The one I mentioned is the closest to the Schrodinger picture of ordinary QM. But it is not very convenient in practice, so practical physicists usually ignore it.

I understand, thanks.

https://www.amazon.com/dp/0201360799/?tag=pfamazon01-20 talks about the Schroedinger picture wave functional formulation (mentioned by @Demystifier), and its equivalence to the more usual Heisenberg picture formulation of QFT.

Well, the book looks ambitious, it says they present string theory assuming you don't know quantum field theory so they explain you in the first part of the book. That has to be very interesting.

 

[DarMM]

WHy is that each one of you gives me a different version?

I will search info about wave functionals.

I appreciate any comment really, thank you so much.

As mentioned above there are many ways of formulating Quantum Field Theory.

The Wavefunctional approach is the form most closely related to the way people normally do non-relativistic QM. However solving for the Wave Functional or Wave Functional eigenvalue problems are virtually impossible outside of free field theories, so people usually use the method of calculating the correlation functions I mentioned. Non-relativistic QM can also be expressed via correlation functions just to be clear.

One thing I didn't mention in my post is renormalization. When we write down QFT we often make use of expressions like $\phi^{3}(x)$. However because the object $\phi(x)$ is so singular, it turns out that $\phi^{3}(x)$ isn't a well defined mathematical expression. So one has to attempt to define a quantity similar to $\phi^{3}(x)$ that is well defined: $[\phi^{3}(x)]_R$.

I'll give an example using plain old functions. Say you have the Heaviside function $\Theta(x)$ and the function $\frac{1}{x}$. We have their multiple $\frac{\Theta(x)}{x}$. However consider the following integral:
$$\int_{\mathbb{R}}{\frac{\Theta(x)}{x}g(x)dx} = \int^{\infty}_{0}{\frac{g(x)}{x}dx}$$
with $g$ some function. The $\Theta$ function has been used to restrict the integral in the second expression.
In general this will diverge if $g$ is non-zero about the point $x = 0$.
However the following modified version of the integral is well-defined:
$$\int^{a}_{0}{\frac{g(x) - g(0)}{x}dx} + \int^{\infty}_{a}{\frac{g(x)}{x}dx}$$
So we isolate the problematic region and add a term that removes the divergence.
This can be rewritten as:
$$\int^{\infty}_{0}{\frac{g(x)}{x}dx} - \int^{a}_{0}{\frac{g(0)}{x}dx}$$
And since $g(0)$ can be obtained from a delta function, we can further rewrite it as:
$$\int^{\infty}_{0}{\frac{g(x)}{x}dx} - \left(\int^{a}_{0}{\frac{1}{x}dx}\right)\int{\delta(x)g(x)dx}$$

Finally if we put back in the Heaviside function, this can be written as:
$$\int_{\mathbb{R}}{\left[\frac{\Theta(x)}{x} - \left(\int^{a}_{0}{\frac{1}{x}dx}\right)\delta(x)\right]g(x)dx}$$

So we see that:
$$\frac{\Theta(x)}{x} - \left(\int^{a}_{0}{\frac{1}{x}dx}\right)\delta(x)$$
is a modified version of:
$$\frac{\Theta(x)}{x}$$
that produces well defined integrals.

The physicist way of doing this is to call
$$\left(\int^{a}_{0}{\frac{1}{x}dx}\right)\delta(x)$$
a counterterm and to consider
$$\left(\int^{a}_{0}{\frac{1}{x}dx}\right)$$
to define an "infinite constant" $C_a$. So they would write:

$$\left[\frac{\Theta(x)}{x}\right]_R = \frac{\Theta(x)}{x} - C_a\delta(x)$$

In physics language you obtain a version of the product of the Heaviside step function and $\frac{1}{x}$ that behaves well under integrals by subtracting off a counterterm which has an infinite constant. This "well behaved" version is called "renormalized".

For some Quantum Field Theories the terms you have to add are no problem. For instance $\phi^{4}$ theory in three dimensions requires:
$$\left[\phi^{4}\right]_R = \phi^{4} - \delta m^{2}\phi^{2}$$

That is making the $\phi^{4}$ interaction well-defined requires adding a counterterm with the $\phi^{2}$ term. However since the Lagrangian already has a $\phi^{2}$ this is no problem, it doesn't change the character of the theory.

However some theories require the addition of new terms to their Lagrangian to be well defined. If this process doesn't stop, i.e. the new terms then need further new terms, the theory is called non-renormalizable.

 

[jonjacson]

As mentioned above there are many ways of formulating Quantum Field Theory.

The Wavefunctional approach is the form most closely related to the way people normally do non-relativistic QM. However solving for the Wave Functional or Wave Functional eigenvalue problems are virtually impossible outside of free field theories, so people usually use the method of calculating the correlation functions I mentioned. Non-relativistic QM can also be expressed via correlation functions just to be clear.

One thing I didn't mention in my post is renormalization. When we write down QFT we often make use of expressions like $\phi^{3}(x)$. However because the object $\phi(x)$ is so singular, it turns out that $\phi^{3}(x)$ isn't a well defined mathematical expression. So one has to attempt to define a quantity similar to $\phi^{3}(x)$ that is well defined: $[\phi^{3}(x)]_R$.

I'll give an example using plain old functions. Say you have the Heaviside function $\Theta(x)$ and the function $\frac{1}{x}$. We have their multiple $\frac{\Theta(x)}{x}$. However consider the following integral:
$$\int_{\mathbb{R}}{\frac{\Theta(x)}{x}g(x)dx} = \int^{\infty}_{0}{\frac{g(x)}{x}dx}$$
with $g$ some function. The $\Theta$ function has been used to restrict the integral in the second expression.
In general this will diverge if $g$ is non-zero about the point $x = 0$.
However the following modified version of the integral is well-defined:
$$\int^{a}_{0}{\frac{g(x) - g(0)}{x}dx} + \int^{\infty}_{a}{\frac{g(x)}{x}dx}$$
So we isolate the problematic region and add a term that removes the divergence.
This can be rewritten as:
$$\int^{\infty}_{0}{\frac{g(x)}{x}dx} - \int^{a}_{0}{\frac{g(0)}{x}dx}$$
And since $g(0)$ can be obtained from a delta function, we can further rewrite it as:
$$\int^{\infty}_{0}{\frac{g(x)}{x}dx} - \left(\int^{a}_{0}{\frac{1}{x}dx}\right)\int{\delta(x)g(x)dx}$$

Finally if we put back in the Heaviside function, this can be written as:
$$\int_{\mathbb{R}}{\left[\frac{\Theta(x)}{x} - \left(\int^{a}_{0}{\frac{1}{x}dx}\right)\delta(x)\right]g(x)dx}$$

So we see that:
$$\frac{\Theta(x)}{x} - \left(\int^{a}_{0}{\frac{1}{x}dx}\right)\delta(x)$$
is a modified version of:
$$\frac{\Theta(x)}{x}$$
that produces well defined integrals.

The physicist way of doing this is to call
$$\left(\int^{a}_{0}{\frac{1}{x}dx}\right)\delta(x)$$
a counterterm and to consider
$$\left(\int^{a}_{0}{\frac{1}{x}dx}\right)$$
to define an "infinite constant" $C_a$. So they would write:

$$\left[\frac{\Theta(x)}{x}\right]_R = \frac{\Theta(x)}{x} - C_a\delta(x)$$

In physics language you obtain a version of the product of the Heaviside step function and $\frac{1}{x}$ that behaves well under integrals by subtracting off a counterterm which has an infinite constant. This "well behaved" version is called "renormalized".

For some Quantum Field Theories the terms you have to add are no problem. For instance $\phi^{4}$ theory in three dimensions requires:
$$\left[\phi^{4}\right]_R = \phi^{4} - \delta m^{2}\phi^{2}$$

That is making the $\phi^{4}$ interaction well-defined requires adding a counterterm with the $\phi^{2}$ term. However since the Lagrangian already has a $\phi^{2}$ this is no problem, it doesn't change the character of the theory.

However some theories require the addition of new terms to their Lagrangian to be well defined. If this process doesn't stop, i.e. the new terms then need further new terms, the theory is called non-renormalizable.

And do we know the physical reason behind a theory being or not being renormalizable?

 

[DarMM]

And do we know the physical reason behind a theory being or not being renormalizable?

I've presented renormalization from a very mathematical point of view, there are other ways of viewing it.

So people approach nonrenormalizibility differently. Some say it simply means the theory isn't well defined, i.e. there's no physical reason as such, it's just that some classic Lagrangians/Hamiltonians that you can write down don't have a well defined quantum version.

Others view all Quantum Field Theories as coming with an intrinsic cut-off, i.e. a maximum length scale at which they can no longer be used due to new physics. In that case non-renormalizible theories are those field theories where the details of the new physics cannot be ignored, you end up with an infinite amount of terms with constants that encapsulate effects from the higher theory, rather than a simple short Lagrangian to describe low energy physics with.

The two views aren't opposed either. You might say a theory doesn't exist because it can only be defined with a cutoff as some approximation of a higher theory, where as if a theory fundamentally exists due to being renormalizable it's because the higher effect don't matter too much to it and it can define a complete (although ultimately incorrect) theory on its own.

 

[A. Neumaier]

https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf

Please explain why this is a response to my comment. Srednicki discusses no lattice approach to QED - only to strongly coupled gauge theories with confinement.

 

[bhobba]

And do we know the physical reason behind a theory being or not being renormalizable?

Yes - in the modern effective field theory approach. Things have moved on since the time Feynman etc worked out renormalization. We now know you can start with just about any QFT, and at low energies the theory will look renormalizable. Such a theory is called an effective field theory. This of course is quite interesting and useful. But if you go to energies that are high enough, renormalizability may break down, and you may not have a quantum field theory at all. So the reason for theories being renormalizable is, at least at the moment, we do not have the technology to reach the energy where it is not.

Wilson sorted it out:
https://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/

There is quite a bit of (advanced) material on EFT to be found on the internet eg:
http://pages.physics.cornell.edu/~ajd268/Notes/EffectiveFieldTheories.pdf

Its actually an interesting area with unexpected phenomena such as triviality occurring. I wish I knew more than I do, and it is an area I am investigating myself.

Thanks
Bill

 

[bhobba]

So people approach nonrenormalizibility differently. Some say it simply means the theory isn't well defined, i.e. there's no physical reason as such, it's just that some classic Lagrangians/Hamiltonians that you can write down don't have a well defined quantum version.

That's true, its the old view of renormalisation. But you have to ask yourself - why is renormalization so fundamental that only such theories are valid? Wilsons view that its simply the low energy limit of a possibly non-renormalizeable theory makes more sense. Of course nature making sense to me may not be how nature works.

Thanks
Bill

 

[A. Neumaier]

people approach nonrenormalizibility differently. Some say it simply means the theory isn't well defined, i.e. there's no physical reason as such, it's just that some classic Lagrangians/Hamiltonians that you can write down don't have a well defined quantum version.

Well, they don't have a unique quantum version. But they have a parameterized family of quantum versions with an infinite number of parameters, most of them being irrelevant except at super high energies. We had discussed this in another thread; see https://www.physicsforums.com/threa...um-field-theory-comments.925220/#post-5844345 and the discussion following that post.