Feynman diagrams and space-time

Hontas Farmer

Happy holidays to all.

I have just finished a formal course based on Peskin and Schroeder's book on
Quantum Field theory. So far the Feynman diagrams have had no explicit
reference to space or time in them. Earlier in my career the Feynman
diagrams I was shown were drawn a certain way with a set of space-time axes
at least implied. The particles proceeding forward through time
anti-particles backward in time, Massless bosons propagating at 45 degree
angles to the axes and such. In this way not simply specifying the
cross-section of a interaction but it's location in space-time.

What is the distinction I should see between a Feynman diagram drawn with
space-time axes implied and without them?
--
Feynman diagrams brought Quantum Field Theory to the Masses. Misha
Stephanov 11/2006. www.geocities.com/hontasfx

sr

Hontas Farmer wrote:

> I have just finished a formal course based on Peskin and Schroeder's book on
> Quantum Field theory. So far the Feynman diagrams have had no explicit
> reference to space or time in them. Earlier in my career the Feynman
> diagrams I was shown were drawn a certain way with a set of space-time axes
> at least implied. The particles proceeding forward through time
> anti-particles backward in time, Massless bosons propagating at 45 degree
> angles to the axes and such. In this way not simply specifying the
> cross-section of a interaction but it's location in space-time.
>
> What is the distinction I should see between a Feynman diagram drawn
> with space-time axes implied and without them?

Feynman diagrams are just pictorial representations for terms in
a perturbation series. Only the topology of the diagram really
matters (i.e: which lines are connected where), as well as the various
momenta directions and vertex factors annotating the diagram.

There's not much point thinking of the internal lines and vertices
as being "located" in space-time. Only the external lines (i.e: the
in-states
and out-states) have any connection to something detectable with
experimental equipment. After all, the whole point of the exercise is
to calculate transition amplitudes between the *asymptotic* in- and
out-states.

HTH.

Igor Khavkine

Hontas Farmer wrote:
> Happy holidays to all.
>
> I have just finished a formal course based on Peskin and Schroeder's book on
> Quantum Field theory. So far the Feynman diagrams have had no explicit
> reference to space or time in them. Earlier in my career the Feynman
> diagrams I was shown were drawn a certain way with a set of space-time axes
> at least implied. The particles proceeding forward through time
> anti-particles backward in time, Massless bosons propagating at 45 degree
> angles to the axes and such. In this way not simply specifying the
> cross-section of a interaction but it's location in space-time.
>
> What is the distinction I should see between a Feynman diagram drawn with
> space-time axes implied and without them?

The short answer is that there is no distinction. Any attempt to endow
internal vertices of Feynman diagrams with time ordering is purely for
illustratory purposes.

The slightly longer answer is that the distinction may have mattered
when at some point, long ago. Namely, Feynman's contribution to
perturbation theory was to make every step of the calculation
relativistically invariant (that would be Poincare invariant). That is
why the diagrams we are all familiar with carry his name. I believe
that diagrams were used to represent individual terms in perturbative
series even before Feynman; unfortunately I don't have a ready
reference. The perturbation calculations used to be done in the
canonical formalism, which required the use of a Hamiltonian and hence
a choice of preferred time direction. In those days, diagrams with all
possible time orientations would have to be considered, hence the
relative ordering of intermediate vertices was important.

The significantly longer answer asks to keep in mind the origin of the
perturbative contributions represented by Feynman diagrams, from which
the importance (or lack thereof) of various space-time labels in an
individual diagram may be deduced.

The first step is to remember that QFT, as any quantum theory, features
states and operators representing physical observables. Expectation
values of these observables and transition probabilities between
different states can be related to experimental measurements. In usual
perturbative QFT, states are provided by Fock space, while all
observables of interest are built out of the field operators from
different space-time points. At this point I'm neglecting both
renormalizability issues as well as the mathematical necessity to treat
fields as operator valued distributions. In this setting, the
calculation of both the expectation values and the transition
amplitudes can be reduced to the evaluation of so-called n-point
correlation functions or Green functions:

G(x_1,x_2,...,x_n) = < phi(x_1) phi(x_2) ... phi(x_n) >,

where <...> denotes the Fock-vacuum expectation value and phi(x_i) is
the Heisenberg-picture field operator at space-time point x_i. In
short, any physically meaningful quantity must be expressible in terms
of these correlation functions.

For simplicity, lets restrict ourselves to two-point functions G(x,y)
only. This correlation function is evaluated by expanding it as a
series in h-bar.

G(x,y) = G_0(x,y) + (h-bar) G_1(x,y) + (h-bar)^2 G_2(x,y) + ... .

Each term G_k(x,y) is represented by a finite number of Feynman
diagrams. The power of h-bar accompanying each term in the expansion
happens to coincide with the number of loops in the corresponding
diagrams.

In QFT books, the above expansion is usually obtained in one of two
ways. One way is the path integral approach, where each term in the
series corresponds to a Gaussian integral. The other is the canonical
approach: write the Hamiltonian as a spatial integral over the
Hamiltonian density, write the time evolution operator in terms of
Dyson's time-ordered exponential, change operators from the Heisenberg
to the interaction picture, represent the interaction-picture
interacting vacuum state as the Heisenberg-picture free vacuum state
evolved from the infinite past/future, form the expectation value and
expand all time-ordered exponentials in powers of h-bar. In fact, the
path-integral and canonical approaches are proved to be equivalent
based on the fact that they give the same answer in these calculations.

Either way, each term G_k(x,y) ends up being a multi-fold space-time
integral. In the path-integral approach, each space-time integral comes
from a factor of the action, whose exponential is being expanded. In
the canonical approach, each space-time integral comes from a factor of
the Hamiltonian (giving the spatial integral) and from the expansion of
the time ordered exponential (giving the temporal integral). The
presence of the space-time integrals is what gives these perturbative
calculations step-by-step relativistic invariance. Using Wick's
theorem, the integrand can be written as a product of free-field
two-point functions, with each of the two arguments being either x, y,
or one of the space-time coordinates being integrated over. Each of the
coordinates (either x, y, or one being integrated over) can be
represented by a dot on a piece of paper. Each free-field two-point
function can be represented by a line connecting the two dots whose
coordinate labels constitute its arguments. This is how Feynman
diagrams are born.

Hontas, if you're still with me, you can now see an answer to your
question. Given a Feynman diagram, there should be a space-time label
on each internal vertex (these are the coordinates being integrated
over) and at the end of each external leg (that would be x or y). I
hope it is by now obvious that the relative position of internal
vertices of a Feynman diagram is quite arbitrary, as each one will
assume all possible space-time coordinates in the course of
integrations necessary to evaluate the diagram. The space-time labels
on the external legs are fixed, and you may, if you wish, position them
on the page keeping some kind of chronological order. But that is by no
means necessary.

The above discussion should also illustrate that a single Feynman
diagram may not mean much on its own. After all, it is only one of the
(sometimes many) terms constituting the expansion coefficient G_k(x,y),
while the latter is one term of the expansion of G(x,y), which
ultimately the quantity of physical interest. However, this sober point
of view in no way detracts from the more poetic illustrative power of
Feynman diagrams.

One last piece to complete the puzzle is the transition from
position-space to momenum-space evaluation of Feynman diagrams. Since
each free-field two-point function, say D(x,y), is translation
invariant, it may be written as D(x-y), that is, as a function of the
single argument x-y. Note that this simplification is impossible in the
presence of some kind of background field or inhomogeneous medium. When
the simplification is possible, the two-point function may also be
expressed in terms of its Fourier transform:

D(x-y) = int D(k) exp[i k.(x-y)] dk .

Once the above identity is substituted into the Feynman diagram
evaluation, the order of momentum and space-time integrals can be
exchanged and the latter evaluated to a bunch of delta-functions. Now,
each line in the diagram is attributed a momentum label that is
integrated over, while the delta-functions enforce "momentum
conservation" at each vertex. The external legs keep some position
dependent polarization factors, since the external coordinate labels
were not integrated over. The change to momentum-space evaluation often
simplifies the calculation since it reduces the number of independent
integrations.

I'm sure that an introductory QFT class has not spent much time
dwelling on a large fraction of the steps described above. However, if
you are intent on attributing physical meaning to Feynman diagrams, you
should be familiar with each of the above steps, since their
understanding is necessary to decide when such attribution is
justified.

Hope this helps.

Igor

Oh No

Thus spake Hontas Farmer <hfarme2@uic.edu>
>Happy holidays to all.
>
>I have just finished a formal course based on Peskin and Schroeder's book on
>Quantum Field theory. So far the Feynman diagrams have had no explicit
>reference to space or time in them. Earlier in my career the Feynman
>diagrams I was shown were drawn a certain way with a set of space-time axes
>at least implied. The particles proceeding forward through time
>anti-particles backward in time, Massless bosons propagating at 45 degree
>angles to the axes and such. In this way not simply specifying the
>cross-section of a interaction but it's location in space-time.
>
>What is the distinction I should see between a Feynman diagram drawn with
>space-time axes implied and without them?

The internal ordering of points in a Feynman diagram is irrelevant. Only
the topology matters. There is certainly no reason to draw lines at
45deg angles, unless it is one of graphical clarity. However, when used
to describe a particular physical interaction, there is a defined in
state and a defined out state, and the external lines will be assigned
either to the in or the out state. Clearly there is a time ordering in
that.



Regards

--
Charles Francis
substitute charles for NotI to email

Hontas Farmer

Igor Khavkine wrote:

> Hontas Farmer wrote:
>> Happy holidays to all.
>>
>> I have just finished a formal course based on Peskin and Schroeder's book
>> on
>> Quantum Field theory. So far the Feynman diagrams have had no explicit
>> reference to space or time in them. Earlier in my career the Feynman
>> diagrams I was shown were drawn a certain way with a set of space-time
>> axes
>> at least implied. The particles proceeding forward through time
>> anti-particles backward in time, Massless bosons propagating at 45 degree
>> angles to the axes and such. In this way not simply specifying the
>> cross-section of a interaction but it's location in space-time.
>>
>> What is the distinction I should see between a Feynman diagram drawn with
>> space-time axes implied and without them?
>
> The short answer is that there is no distinction. Any attempt to endow
> internal vertices of Feynman diagrams with time ordering is purely for
> illustratory purposes.
>
> The slightly longer answer is that the distinction may have mattered
> when at some point, long ago. Namely, Feynman's contribution to
> perturbation theory was to make every step of the calculation
> relativistically invariant (that would be Poincare invariant). That is
> why the diagrams we are all familiar with carry his name. I believe
> that diagrams were used to represent individual terms in perturbative
> series even before Feynman; unfortunately I don't have a ready
> reference. The perturbation calculations used to be done in the
> canonical formalism, which required the use of a Hamiltonian and hence
> a choice of preferred time direction. In those days, diagrams with all
> possible time orientations would have to be considered, hence the
> relative ordering of intermediate vertices was important.

Ah so I am not crazy. The diagrams you may be thinking of are diagrams that
are used in classical statistical physics.

>
> The significantly longer answer asks to keep in mind the origin of the
> perturbative contributions represented by Feynman diagrams, from which
> the importance (or lack thereof) of various space-time labels in an
> individual diagram may be deduced.
>
> The first step is to remember that QFT, as any quantum theory, features
> states and operators representing physical observables. Expectation
> values of these observables and transition probabilities between
> different states can be related to experimental measurements. In usual
> perturbative QFT, states are provided by Fock space, while all
> observables of interest are built out of the field operators from
> different space-time points. At this point I'm neglecting both
> renormalizability issues as well as the mathematical necessity to treat
> fields as operator valued distributions. In this setting, the
> calculation of both the expectation values and the transition
> amplitudes can be reduced to the evaluation of so-called n-point
> correlation functions or Green functions:
>
> G(x_1,x_2,...,x_n) = < phi(x_1) phi(x_2) ... phi(x_n) >,
>
> where <...> denotes the Fock-vacuum expectation value and phi(x_i) is
> the Heisenberg-picture field operator at space-time point x_i. In
> short, any physically meaningful quantity must be expressible in terms
> of these correlation functions.
>
> For simplicity, lets restrict ourselves to two-point functions G(x,y)
> only. This correlation function is evaluated by expanding it as a
> series in h-bar.
>
> G(x,y) = G_0(x,y) + (h-bar) G_1(x,y) + (h-bar)^2 G_2(x,y) + ... .
>
> Each term G_k(x,y) is represented by a finite number of Feynman
> diagrams. The power of h-bar accompanying each term in the expansion
> happens to coincide with the number of loops in the corresponding
> diagrams.
>
> In QFT books, the above expansion is usually obtained in one of two
> ways. One way is the path integral approach, where each term in the
> series corresponds to a Gaussian integral. The other is the canonical
> approach: write the Hamiltonian as a spatial integral over the
> Hamiltonian density, write the time evolution operator in terms of
> Dyson's time-ordered exponential, change operators from the Heisenberg
> to the interaction picture, represent the interaction-picture
> interacting vacuum state as the Heisenberg-picture free vacuum state
> evolved from the infinite past/future, form the expectation value and
> expand all time-ordered exponentials in powers of h-bar. In fact, the
> path-integral and canonical approaches are proved to be equivalent
> based on the fact that they give the same answer in these calculations.
>
> Either way, each term G_k(x,y) ends up being a multi-fold space-time
> integral. In the path-integral approach, each space-time integral comes
> from a factor of the action, whose exponential is being expanded. In
> the canonical approach, each space-time integral comes from a factor of
> the Hamiltonian (giving the spatial integral) and from the expansion of
> the time ordered exponential (giving the temporal integral). The
> presence of the space-time integrals is what gives these perturbative
> calculations step-by-step relativistic invariance. Using Wick's
> theorem, the integrand can be written as a product of free-field
> two-point functions, with each of the two arguments being either x, y,
> or one of the space-time coordinates being integrated over. Each of the
> coordinates (either x, y, or one being integrated over) can be
> represented by a dot on a piece of paper. Each free-field two-point
> function can be represented by a line connecting the two dots whose
> coordinate labels constitute its arguments. This is how Feynman
> diagrams are born.
>
> Hontas, if you're still with me, you can now see an answer to your
> question. Given a Feynman diagram, there should be a space-time label
> on each internal vertex (these are the coordinates being integrated
> over) and at the end of each external leg (that would be x or y). I
> hope it is by now obvious that the relative position of internal
> vertices of a Feynman diagram is quite arbitrary, as each one will
> assume all possible space-time coordinates in the course of
> integrations necessary to evaluate the diagram. The space-time labels
> on the external legs are fixed, and you may, if you wish, position them
> on the page keeping some kind of chronological order. But that is by no
> means necessary.

In a short over simplified way, you are saying that any space-time
dependence is being integrated out of the calculation at every vertex.

>
> The above discussion should also illustrate that a single Feynman
> diagram may not mean much on its own. After all, it is only one of the
> (sometimes many) terms constituting the expansion coefficient G_k(x,y),
> while the latter is one term of the expansion of G(x,y), which
> ultimately the quantity of physical interest. However, this sober point
> of view in no way detracts from the more poetic illustrative power of
> Feynman diagrams.
>
> One last piece to complete the puzzle is the transition from
> position-space to momenum-space evaluation of Feynman diagrams. Since
> each free-field two-point function, say D(x,y), is translation
> invariant, it may be written as D(x-y), that is, as a function of the
> single argument x-y. Note that this simplification is impossible in the
> presence of some kind of background field or inhomogeneous medium. When
> the simplification is possible, the two-point function may also be
> expressed in terms of its Fourier transform:
>
> D(x-y) = int D(k) exp[i k.(x-y)] dk .
>
> Once the above identity is substituted into the Feynman diagram
> evaluation, the order of momentum and space-time integrals can be
> exchanged and the latter evaluated to a bunch of delta-functions. Now,
> each line in the diagram is attributed a momentum label that is
> integrated over, while the delta-functions enforce "momentum
> conservation" at each vertex. The external legs keep some position
> dependent polarization factors, since the external coordinate labels
> were not integrated over. The change to momentum-space evaluation often
> simplifies the calculation since it reduces the number of independent
> integrations.
>
> I'm sure that an introductory QFT class has not spent much time
> dwelling on a large fraction of the steps described above. However, if
> you are intent on attributing physical meaning to Feynman diagrams, you
> should be familiar with each of the above steps, since their
> understanding is necessary to decide when such attribution is
> justified.
>
> Hope this helps.
>
> Igor

Yes that helps very much. As the professor put it it could take many years
to fully understand this topic. To be honest I am going to have to reread
that reply several times.

Oh No has written:
>The internal ordering of points in a Feynman diagram is irrelevant. Only
>the topology matters. There is certainly no reason to draw lines at
>45deg angles, unless it is one of graphical clarity. However, when used
>to describe a particular physical interaction, there is a defined in
>state and a defined out state, and the external lines will be assigned
>either to the in or the out state. Clearly there is a time ordering in
>that.

It could be that the diagrams that I was shown before taking this course in
QFT were based on standard Feynman diagram formalism but not 100% standard.
The rationale for drawing the lines certain ways and making reference to
space-time was as follows. A massless Boson (or fermion for that matter)
would travel at the speed of light. If space and time units are equal...as
in special relativity..then such particles could only propagate with the
speed of light. All other material particles would travel at some other
sub-light speed. Antiparticles would be drawn such that they would
approach from a latter time to an earlier time. Particles would seem to
travel forwards in time. Basically the space-time axes and these concepts
I have described were used by this one particular professor to denote just
what type of particle a given particle was. For the record he NEVER said
that any crossections depended on space-time coordinates. We just used
then to discuss particle interactions. Questions like..."Can a photon of
sufficient energy decay into an electron and a positron? Why or why not?
and such"

Thanks for all your help.

--
Feynman diagrams brought Quantum Field Theory to the Masses. Misha
Stephanov 11/2006

Igor Khavkine

Hontas Farmer wrote:
> Igor Khavkine wrote:

[...]
> In a short over simplified way, you are saying that any space-time
> dependence is being integrated out of the calculation at every vertex.

That's a good way to put it.

[...]
> > Hope this helps.

> Yes that helps very much. As the professor put it it could take many years
> to fully understand this topic. To be honest I am going to have to reread
> that reply several times.

That's a good idea. I certainly didn't grasp all that I talkd about
over night. If you're serious about it, I can give textbook references
for most of the steps I described (sorry, to my knowledge, no one
textbook would be sufficient). If you search for my past posts about
Feynman diagrams, you'll find that I've already cited some of them
before.

Igor

Oh No

Thus spake Hontas Farmer <hfarme2@uic.edu>
>Igor Khavkine wrote:
>
>>
>> The slightly longer answer is that the distinction may have mattered
>> when at some point, long ago. Namely, Feynman's contribution to
>> perturbation theory was to make every step of the calculation
>> relativistically invariant (that would be Poincare invariant). That is
>> why the diagrams we are all familiar with carry his name. I believe
>> that diagrams were used to represent individual terms in perturbative
>> series even before Feynman; unfortunately I don't have a ready
>> reference. The perturbation calculations used to be done in the
>> canonical formalism, which required the use of a Hamiltonian and hence
>> a choice of preferred time direction. In those days, diagrams with all
>> possible time orientations would have to be considered, hence the
>> relative ordering of intermediate vertices was important.
>
>Ah so I am not crazy. The diagrams you may be thinking of are diagrams that
>are used in classical statistical physics.

That sounds probable, though I am not sufficiently familiar with
statistical physics to say. I think it was almost exclusively Feynman
who thought that playing with these diagrams could lead to an advance in
understanding, but the calculation which removes time ordering from the
diagrams and shows them to be equivalent to the canonical formulation
was actually done by Dyson, on his famous bus journey (famous for this
calculation).
>
>In a short over simplified way, you are saying that any space-time
>dependence is being integrated out of the calculation at every vertex.

Precisely. That is the basis of the calculation. One thing to watch out
for - swapping the order of integration isn't strictly legitimate in
rigorous analysis. This is called "the abuse of Wick's theorem" by
Scharf in Finite Quantum electrodynamics and can be regarded as the
origin of the ultraviolet divergence. I think once this is understood,
Feynman diagrams can also be understood as having much more physical
meaning than is often said.

>Oh No has written:
>>The internal ordering of points in a Feynman diagram is irrelevant. Only
>>the topology matters. There is certainly no reason to draw lines at
>>45deg angles, unless it is one of graphical clarity. However, when used
>>to describe a particular physical interaction, there is a defined in
>>state and a defined out state, and the external lines will be assigned
>>either to the in or the out state. Clearly there is a time ordering in
>>that.
>
>It could be that the diagrams that I was shown before taking this course in
>QFT were based on standard Feynman diagram formalism but not 100% standard.
>The rationale for drawing the lines certain ways and making reference to
>space-time was as follows. A massless Boson (or fermion for that matter)
>would travel at the speed of light. If space and time units are equal...as
>in special relativity..then such particles could only propagate with the
>speed of light. All other material particles would travel at some other
>sub-light speed. Antiparticles would be drawn such that they would
>approach from a latter time to an earlier time. Particles would seem to
>travel forwards in time.

This isn't actually strictly true. There is a non-zero amplitude for the
creation of a particle and its annihilation outside the light cone, even
for a massive particle. This is an unobservable process, because it is
always matched by the corresponding antiparticle going the other way,
and the amplitude for both processes taken together cancels to zero.
Each line in a Feynman diagram incorporates the amplitudes for both the
particle and the antiparticle.
>

Regards

--
Charles Francis
substitute charles for NotI to email

Hontas Farmer

Happy Gregorian New year to you all!

I have been thinking over what was said about the presence of space-time on
the Feynman diagrams that I saw earlier... In condensed form, space-time
coordinates are summed over and the final amplitude does not depend on
them. The same goes for these diagrams when drawn in momentum space.
(Personally I find momentum space easier to work with. There is no reason
to insert what amounts to a Fourier transform into the process. Among
other things.) I get where you are all coming from on that.

So to get a better idea of what was going on I decided to reread "QED: The
Strange Theory of Light and Matter" by Richard P. Feynman. In this book
which is based heavily on a series of lectures on QED he gave he works his
way up to drawing Feynman diagrams as a way of calculating the possible
outcomes of various particles interacting. For a good portion of the book
he explicitly draws in Space-Time axes and does not trivialize the
importance of space-time. In particular the fact that this space time is
presumed to be a non-reactive spectator to what is going on.

I have scanned some figures I would like you all to see.

<a href="http://photobucket.com/" target="_blank"><img
src="http://i57.photobucket.com/albums/g204/Zahara_TS/fyenmanhadST.jpg"
border="0" alt="Photobucket - Video and Image Hosting"></a>

In the whole thing he alludes in as un-mathematical a way as possible to the
fact that these diagrams are used in a perturbation series. But that does
not change the fact that space-time is present for all of this.

OK. Basically...this is what I get from reading this book right through.
He is saying that to compute the full coruscation of an interaction one
needs to draw diagrams for every way the interaction can occur. Every way
that is allowed to connect external lines, and vertices's with the given
set of particle interactions. Then add up the results.

That as we all know is how one finds the quantity that most modern particle
physicist are interested in. The cross section (units of area) that
expresses the probability of a given interaction.

THERE IS HOWEVER ONE OTHER QUANTITY WHICH DR. FEYNMAN MENTIONED
Often...the "Amplitude".

Each arrow on the diagram has a length called it's amplitude (not unlike in
classical physics, in fact quite analogous). He describes the process as
that of calculating the "final arrow" for the event. It's probability
amplitude.

Does this difference in what was being calculated (the cross-section of a
vertex vs the amplitude of an arrow) effect the answers I got before? It
seems to and I am racking my brain to figure out how.

Any help will be appreciated and if I am able I will help you someday.

Thank you

--
Feynman diagrams brought Quantum Field Theory to the Masses. Misha
Stephanov 11/2006