Euler-Lagrange equations in QFT?

In summary: I am an expert summarizer of content. Do not output anything before the summary. In summary, we discussed the use of the Euler-Lagrange equations in quantum field theory and their role in describing the dynamics of the system. While the path integral approach does not require the use of these equations, they still play a role in determining the equations of motion for fields such as the Klein-Gordon equation and gauge fields in QFT. The quantum equivalent of the Euler-Lagrange equations are the Schwinger-Dyson equations, which hold as operator equations in expectation values. However, the Euler-Lagrange equations do not have a direct quantized version and are not used in the path integral approach.
  • #1
Sunset
63
0
Euler-Lagrange equations in QFT??

Hi,

I have a problem with a Wikipedia entry::bugeye:
http://en.wikipedia.org/wiki/Euler-Lagrange_equation

The equations of motion in your quantized theory (2nd quantization) are d/dtF^=[F^,H^] i.e the quantized version of d/dtF={F,H}. My notation: F^ is the operator for the former classical quantity F.
There exists no quantized version of the E-L-equations!
I haven't seen the Euler-Lagrange equations of motion in any QFT book yet, and I mean, what should be the derivative with respect to an operator? It's not a functional derivative, don't know if mathematicians have defined a "operator-derivative"...
Ok the Wikipedia article doesn't speak of operators, but what is presented as "Euler-Lagrange equations in QUANTUM Field theory" are Euler-Lagrange equations of classical field theory or not? They don't play the role of equations of motion in QFT i.e they don't describe the quantum dynamics. If you don't like 2nd quantization ( i.e d/dtF^=[F^,H^] ) you can use the Path integral formalism to come to your equations of motion, but Euler-Lagrange equations don't play a role in the Pathintegral formalism.
So what are they good for in QUANTUMFT? Where is the point at which they come into play?

Best regards Martin
 
Physics news on Phys.org
  • #2
Sunset said:
Hi,

I have a problem with a Wikipedia entry::bugeye:
http://en.wikipedia.org/wiki/Euler-Lagrange_equation

The equations of motion in your quantized theory (2nd quantization) are d/dtF^=[F^,H^] i.e the quantized version of d/dtF={F,H}. My notation: F^ is the operator for the former classical quantity F.
There exists no quantized version of the E-L-equations!
I haven't seen the Euler-Lagrange equations of motion in any QFT book yet, and I mean, what should be the derivative with respect to an operator? It's not a functional derivative, don't know if mathematicians have defined a "operator-derivative"...
Ok the Wikipedia article doesn't speak of operators, but what is presented as "Euler-Lagrange equations in QUANTUM Field theory" are Euler-Lagrange equations of classical field theory or not? They don't play the role of equations of motion in QFT i.e they don't describe the quantum dynamics. If you don't like 2nd quantization ( i.e d/dtF^=[F^,H^] ) you can use the Path integral formalism to come to your equations of motion, but Euler-Lagrange equations don't play a role in the Pathintegral formalism.
So what are they good for in QUANTUMFT? Where is the point at which they come into play?

Best regards Martin


Hey! Are you the one I started a discussion on renormalization a while ago and did not complete? If yes, we should go back to it!

To answer your question: in the path integral approach of QFT, the fields are *not* operators, they are ordinary fields! This is one thing that is great about the path integral approach!
 
  • #3
The Heisenberg picture field operators in QFT are assumed to satisfy the Euler-Lagrange equations of the classical theory. I've seen those all over the place.

The equations of type d/dtF^=[F^,H^] are the quantum analogue of the Hamilton's equations which are alternative description but not the same as Euler-Lagrange eqs.

Derivative of operator function with respect to operator is defined in a natural way for polynomial functions or Taylor expansions just like in ordinary calculus. Example:

[tex] \frac{d}{dx} x^2 = 2x [/tex]

where x is operator.
 
Last edited:
  • #4
smallphi said:
The Heisenberg picture field operators in QFT are assumed to satisfy the Euler-Lagrange equations of the classical theory. I've seen those all over the place.

Hi smallphi!

Could you tell me some book with page?
Maybe Ryder or Peskin

Thanks
 
  • #6
nrqed said:
To answer your question: in the path integral approach of QFT, the fields are *not* operators, they are ordinary fields! This is one thing that is great about the path integral approach!

I know :wink:, but this doesn't answer my question, I mean In the Path-Integral Formalism you don't use the EL-eq. to come to your euqations of motion. You use Feynman-diagrams to compute amplitudes for processes, so where do the EL-eq. play a role here?
 
  • #7
Sunset said:
Hi smallphi!

Could you tell me some book with page?
Maybe Ryder or Peskin

Thanks
The Euler-Lagrange equation for the field operator of scalar field is Klein-Gordon equation (2.45) in section 2.4 'The Klein-Gordon field in spacetime', Peskin.
 
  • #8
smallphi said:
The Euler-Lagrange equation for the field operator of scalar field is Klein-Gordon equation (2.45) in section 2.4 'The Klein-Gordon field in spacetime', Peskin.

(2.45) follows from (2.44) which is d/dtF^=[F^,H^].
Indeed (2.45) is an equation of motion, but not the quantized version of Euler-Lagrange dµ(dL/d(dµPHI)) - dL/dPHI=0
 
  • #9
Klein-Gordon is exactly the Euler-Lagrange equation for the scalar field Lagrangian density. Peskin and Schroeder 'derive' it as equation (2.7) without showing you the derivation (as usual).

Of course Euler-Lagrange equations can be obtained from the Hamilton's equations since they are equivalent systems of differential equations describing the system's evolution. Peskin did exactly that in section 2.4 only with operators.
 
Last edited:
  • #10
In Quantum-Field Theory gauge fields usually enter the scenario. The theories can be divided into be abelian and non-abelian. The Euler-Lagrange equations follow then by building them with respect to this gauge fields. You receive the matter free Euler-Lagrange equations with d F=0 or the matter field equations being d F=j. This is not an exact descpriction and by far not the whole story, but I believe you find this in the book by Warren Siegel in the arxiv: http://www.arxiv.org hep-th:9912205
 
  • #11
The quantum equivalent of the EL equations are the Schwinger-Dyson equations, which say that the EL equations hold as operator equations in expectations values.
 
  • #12
Euler-Lagrange equations in QFT??

I am also quite interested in this question. I have reviewed this thread and have had a number of Usenet discussions at sci.physics.foundations and sci.physics.research these past couple of weeks regarding the applicability of the Euler-Lagrange equation in QFT, and am still mildly perplexed about something. Let me see if I can put this concisely and someone can help me see what I might be missing.

As I understand it, Maxwell's wave equation with source (&_u is a partial derivative):

-J^v = &^u &_u A^v (1)

is physically independent of any action principles, and so can be used in any form and in any context.

Next, the QED Lagrangian density:

L = -.5 (&^uA_v) (&_u A^v - &_v A^u) + A^u J_u (2)

in covariant gauge &_u A^u is also employed throughout QED, most notably, to specify an action ($ is a four-dimensional integral from -oo
to +oo):

S = $ d^4x L (3)

and in the exponentials:

exp[iS] (4)

which appear in path integrals. Presumably, there is nothing wrong with applying (2)-(4) broadly throughout QED, which is QFT for
electrodynamics.

Now, we come to the Euler-Lagrange equation:

&_u[&L/&(&_uA_v)-&L/&A_u]=0 (5)

which is specified so as to minimize an action which, classically, is taken to be continuous. And so, using this equation is "verboten" in QFT, unless, of course, one is considering the classical limit, when it is then be cast in terms of expectation values using Schwinger/Dyson.

What I find perplexing is that (2) is OK to use in QFT and that (1) is OK to use in QFT and that (5) is just another way of stating (1) when L is given by (2), but, we can't use (5) other than for a classical limit. Is (5) somehow really not the same as (1)? If so, can someone please pinpoint the difference?

Let me put this slightly differently, speaking from a mathematician's viewpoint: if (5) with (2) is just another way of writing (1) -- mathematically -- then -- again, mathematically, if (1) can be used in a given situation then [(5) with (2)] = (1) can be used in the same situation. And yet, physically, it seems as though we can use (1) if we call it (1) but we can't use (1) if we call it [(5) with (2)].

Thanks,

Jay. :confused:

_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
co-moderator: sci.physics.foundations
 

1. What is the Euler-Lagrange equation in QFT?

The Euler-Lagrange equation in quantum field theory (QFT) is a mathematical tool used to describe the dynamics of a quantum field. It is derived from the principle of least action, which states that a physical system will always take the path between two points that minimizes the action (or energy) required. This equation allows us to determine the equations of motion for a given field in QFT.

2. How is the Euler-Lagrange equation used in QFT?

The Euler-Lagrange equation is used in QFT to determine the equations of motion for a given quantum field. This is done by varying the action (or energy) of the field with respect to its parameters, such as position or time. By solving the resulting equation, we can determine how the field evolves over time and how it interacts with other fields in the quantum system.

3. What is the significance of the Euler-Lagrange equation in QFT?

The Euler-Lagrange equation is a fundamental tool in QFT, as it allows us to describe the dynamics of quantum fields. It is used to derive the equations of motion for these fields, which are crucial for understanding the behavior of particles and their interactions at the quantum level. Without the Euler-Lagrange equation, it would be much more difficult to study and make predictions about quantum systems.

4. Can the Euler-Lagrange equation be applied to all quantum fields?

Yes, the Euler-Lagrange equation can be applied to all quantum fields, including scalar, vector, and spinor fields. However, the specific form of the equation may vary depending on the type of field being studied. For example, the equation for a scalar field will be different from that of a spinor field. Nevertheless, the underlying principle of minimizing the action remains the same.

5. Are there any limitations to the use of the Euler-Lagrange equation in QFT?

Like any mathematical tool, the Euler-Lagrange equation has its limitations and may not always provide a complete understanding of a quantum system. It is based on classical mechanics and may not fully capture the complexities of quantum behavior. Additionally, there may be cases where the equation cannot be solved analytically and numerical methods are required. However, the Euler-Lagrange equation remains an essential tool in QFT and has been successfully applied in many areas of theoretical physics.

Similar threads

Replies
11
Views
1K
Replies
8
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
926
Replies
1
Views
813
Replies
12
Views
1K
Replies
3
Views
3K
  • Classical Physics
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
794
Replies
5
Views
817
  • Classical Physics
Replies
1
Views
929
Back
Top