Ув. shkrobius обратил мое внимание на то, что фраза в заголовке моего дневника принадлежит Д. Мермину, а не В. Паули (как думал я) или Р. Фейнману (как думают многие другие). Так что я поменял его на подлинный совет Фейнмана математическим физикам.
Ага, вот это и есть граница между математиками и всеми остальными специалистами. Которые верят в этот совет и следуют ему, те не математики. Математики всегда стремятся к строгости, и строгость им помогает, а не мешает.
Ты с Максимом К. много о математике беседовал? А с Борей Ф.? Я хочу сказать, что можно девиз сей понимать не в смысле конечного результата, а в смысле полёта мысли. Каковой полёт у каждого математика происходит по своим законам. У кого-то -- строго, у кого-то -- в облаке нечётких формулировок. Итоговый результат вполне мозет быть строгим.
Полет мысли у всех происходит по разному, и, думаю, у всех в той или иной нечеткой форме. В моем комменте не утверждается обратного, а утверждается то, что в нем сформулировано. С М.К. и Б.Ф. я беседовал и они стремятся к строгости, да, как и все математики. Сила этого стремления может быть разной в зависимости -- не столько даже от математика, сколько от пары (математик, обдумываемый вопрос). Если у кого-то по какому-то вопросу много интуиции, ему достаточно меньшей строгости, если мало, нужно больше строгости. Но знак этого стремления всегда положительный, а не отрицательный. Я об этом. Поскольку лично мои утверждения для упомянутых М.К. и Б.Ф. не очень интуитивны, они спрашивают у меня доказательств. И мне тоже удалось извлечь из М.К. одно вполне себе доказательство, чему свидетельством моя статья (хотя сначала он говорил какие-то очень мутные для меня слова).
Не говоря о том, что я лично наблюдал, как М.К. вкладывал вполне ощутимые усилия во вполне технический вопрос об определении понятий единицы и тождественного морфизма в А_бесконечность алгебрах и категориях. И воспринимал его нерешенность как серьезное препятствие в своей работе про А_бесконечность категории. Каковой вопрос он и решил успешно, придумав строгие и гомотопические единицы (доказательство его утверждения об их эквивалентности было потом прописано в диссертации Лефевра-Хасегавы).
Я думаю, ты не принимаешь во внимание специфику математической физики. Предмет там задан "свыше" (допустим, КТП), но аксиомы и основные определения не совсем понятны. Чтобы их "найти", надо сначала проделать много нестрогих вычислений, чтобы "прочувствовать" предмет. Попытка делать строго все с самого начала приведет, скорее всего, к ступору. Ну, или к неправильному набору аксиом, что еще хуже, поскольку создает иллюзию понимания там, где ее нет.
Если область старая, то все основные определения уже известны. Если область совсем новая, то кто-то должен сделать предварительную работу по продумыванию определений, и пока эта работа происходит, о строгости можно говорить только как о конечной цели. Иногда математики сами проделывают такую предварительную работу (см. историю анализа), но иногда за них это делают другие, а математики потом приходят "на готовенькое".
"... кто-то должен сделать предварительную работу по продумыванию определений, и пока эта работа происходит, о строгости можно говорить только как о конечной цели."
Например, в алгебраической геометрии после Зарисского-ван дер Вардена-Вейля все было строго, а определения менялись, и совсем не ясно, что текущие определения - навсегда. Эта работа по продумыванию определений - не "предварительная". Равно она была не "предварительной" и в истории анализа. Анализ был до завершения (на данный момент) работы по продумыванию определений функций, рядов и т.п. и продолжает существовать и после, когда эти определения уже установились. Продумывание и поиск правильных определений - главное в развитии математики.
Есть у меня гипотеза, что слѣдованіе совѣту "замолчи и вычисляй" привело къ тому, что мы такъ до сихъ поръ и не поняли, что именно вычисляемъ въ квантовой теоріи поля, когда дѣлаемъ "перенормировку". "Перенормировка" - это математически некорректная операція, поэтому мы просто не знаемъ, что за отвѣтъ она даётъ. Вычисляемъ - да, а вотъ что именно вычислили (какую математическую величину) - этого не знаемъ.
По-моему надъ этимъ вопросомъ сейчасъ почти никто вообще не работаетъ. А вопросъ важный, т.к. мы вообще не имѣемъ теоріи по-настоящему, пока не понимаемъ этого вопроса.
Напримѣръ, есть разные методы "перенормировки" (размѣрная, дзета-функція, Паули-Вилларсъ, и т.д.). Для нѣкоторыхъ теорій эти методы даютъ одинаковый отвѣтъ. Но Если бы разные методы "перенормировки" давали бы разные отвѣты, не согласующіеся съ экспериментомъ, то это вѣдь не означало бы, что теорія невѣрна - это означало бы, что мы просто не знаемъ, что дѣлаетъ данный методъ "перенормировки". Можетъ, есть какой-то другой методъ, дающій правильный отвѣтъ. Согласіе же отвѣта съ экспериментомъ опять-таки не значитъ, что методъ "перенормировки" правильный: есть много способовъ получать правильный отвѣтъ невѣрными вычисленіями. Пока мы не знаемъ, какой методъ по-настоящему правильный, мы не понимаемъ, что мы вычисляемъ.
(I answer in English because I am not familiar with the RG terminology in Russian).
I think this problem has been resolved quite satisfactorily in the 1960s, thanks to the works of Ken Wilson and his notion of effective action. I also think you are mixing two different things: regularization and renormalization. The 1st key point is that one should always think about QFT with a UV cutoff. Then there are no infinities, but there is a dependence on the details of the cut-off. The cut-off dependence is of two kinds: power-like and logarithmic. The 2nd key point is that this dependence can be compensated by the redefinition of the parameters of the Lagrangian. The 3rd key point is that logarithmic dependence of these parameters is controlled by the RG equations whose form can be computed with any cut-off. Different methods of regularization correspond to different choices of the cut-off, but it is a theorem that the resulting RG equations are independent of the choice, up to a redefinition of the coupling constants. Physical quantities (like S-matrix elements) therefore can be computed using any regularization method, and the results are guaranteed to be related by a redefinition of the coupling constants.
Yes, I said "renormalization method" where I should have said "regularization scheme" to be precise.
It seems to me that the problem has not been solved satisfactorily because things stand as follows with QFT (please correct me if I am wrong here):
1. QFT is formulated as a theory with a cutoff parameter. If I am to take this seriously then I must consider a Lagrangian with coupling constants that are functions of a parameter, which is not a measurable value but a theoretically introduced unknown number. Then I start computing something in perturbation theory using this Lagrangian. Sure enough, I get infinities. Then I realize that I need to cut off something. Mysteriously (and without mathematical justification), the same unknown number is to be used as a cutoff parameter in my perturbative calculations, so that I get finite answers instead of infinities. How can it be that the unknown parameter of the Lagrangian is the same number as the cutoff introduced by hand into some integral in the middle of my calculation? To me, this procedure just never made any sense. I can memorize this procedure, but I cannot understand what is being done. There must be an explanation of what we are calculating here, and then there will remain a technical question of how to calculate it.
2. There exist theories where actual results in perturbation theory (matrix elements or beta functions) do depend on the regularization scheme. Not in QED, but in some QCD calculations and also in some mock-ups of quantum gravity.
1. No, it is not like this. The Lagrangian without a cut-off is not a well-defined starting point for doing quantum computations. Therefore one should start with a Lagrangian together with a cut-off. One DOES NOT introduce the cut-off in the middle, as an afterthought. The parameters of the theory are the parameters in the Lagrangian and the cut-off. These parameters are not measurable. The RG semigroup is a reflection of this: it is a kind of "gauge-invariance", in the sense that all observable quantities are RG-invariant. Physical parameters are labeled by orbits of the RG action on the space of parameters (including the cut-off). It is convenient to work with unmeasurable quantities at intermediate stages of the computation, just like it is convenient to work with vector potential in gauge theory, even though it is not physical. Picking a particular cut-off procedure is like picking a gauge. Of course, it would be even nicer to learn how to do computations without picking a gauge/cut-off procedure. It is not known how to formulate everything in terms of gauge-invariant quantities alone in the gauge theory, and likewise it is not known how to do computations in terms of RG-invariant quantities alone, that is true. But this is an aesthetic problem, and most people do not regard is as a problem at all.
2. (a) The numerical values of the beta-functions are not physical. Beta-functions are components of a vector field on the space of coupling constants. Choosing a different renormalization scheme results in a reparameterization of the coupling constants, which in turns changes the numerical values of the components of the vector field. This does not mean that physical results are scheme-dependent. The vector field is still the same. (b) Physical S-matrix elements are scheme-independent too. One sometimes decomposes them into unphysical "constituents". For example, one writes a hadron scattering cross-section in terms of parton distributions and parton scattering cross-sections. Parton distributions and cross-sections are scheme-dependent, but the hadron cross-section is scheme-independent.
If I understood you correctly, the consistent procedure is that one has to choose three things at once: a) a Lagrangian with coupling constants depending on the cutoff parameter in a certain way, b) a particular regularization scheme that will be later used for calculations, in which the same cutoff parameter also enters, c) the value of the cutoff parameter.
Then one performs calculations and finds that the physical results are invariant under RG flow.
The RG flow corresponds to changing the value of the cutoff parameter, after which the objects a), b) also change simultaneously, so that the physical results stay the same.
I still have two uneasy questions about this:
1) We now have several specific methods of regularization: zeta-function, dimensional regularization, Pauli-Villars, and a few more perhaps that I can't remember now. We can perhaps prove the properties of the RG flow when these methods are used. How can one prove that any method of regularization invented in the future will still yield the same results?
2) If we change the method of regularization, e.g. if we change from zeta-function to dimensional regularization, then the functional dependence of the Lagrangian on the cutoff parameter will be different. It is not equivalent to changing the value of the cutoff parameter (this would be indeed just a different "gauge"). So, as far as I understand, the Lagrangian (a) needs to know in advance that we are using a particular regularization scheme (b) in the middle of our calculations. Or else there will be no cancellation of infinities.
If this is true then it appears that the theory is not really there. Imagine that we are doing classical electrodynamics where we need to solve a wave equation. We have two methods of solving the wave equation: say, by Fourier transform and by separation of variables. Of course, the results are the same: these are just two different techniques for solving a well-defined mathematical problem. Now, imagine that the results are incorrect unless you need to keep two different Lagrangians for the electromagnetic field: one Lagrangian is to be used when we solve the wave equation through the Fourier transformation, but a different Lagrangian is to be used when we solve the wave equation through separation of variables. The two Lagrangians are not equivalent and not simply a redefinition for convenience; these are two really different Lagrangians, with different coupling constants and maybe different terms.
Surely if this were the situation in classical electrodynamics, there would have been a major lack of understanding. If we really understand what we are calculating, then we should start with an initial well-defined mathematical object (e.g. a wave equation) and calculate something. It seems that in QFT we do not have such an initial mathematically well-defined object.
This is almost right. I would put it this way. You need to do either (a) and (b), or (b) and (c). If you do (b) and (c), i.e. fix a specific value of the cut-off, you do not need to regard coupling constants as functions of the cut-off, because the cut-off is fixed. The coupling constants become ordinary numbers. If you do (a) and (b), then you do not need to fix any particular value of the cut-off, since the dependence on it will cancel in the end anyway.
(1) The answer to the first question is that all methods of regularization, implicitly or explicitly, consist of modifying the theory in the UV by adding operators of arbitrarily high dimension to the action. That is, the difference between any two methods of regularization is the addition of such terms. Their effect on the low-energy physics (i.e. on the renormalizable terms in the low-energy effective action) is at most a finite change in the values of the dimensionless coupling constants and masses. Hence all methods are equivalent.
(2) The Lagrangians used with two different regularization schemes are the same. The only difference is that the functional dependence of the coupling constants on the cut-off is different. So the Lagrangians are related in a simple way: by a cut-off-dependent reparameterization of the coupling constants. There is a well-defined procedure how to express one set of coupling constants in terms of another one. How is this different, for example, from transforming one classical action to another one by solving equations of motion for some of the fields and plugging the solutions back in? The new action will look different, but it is still equivalent to the old one.
Well, you see, I have been trying to understand this for a long time, and I never really succeeded. There is no book that I've seen where things are explained straightforwardly.
1) For example, there is the reasoning about UV modifications: one says that by adding operators to the Lagrangian the effect on the low-energy effective theory will be such and such. This would be a valid argument if we already had something well-defined that we are calculating, but we don't, and we are trying to use this very argument to show that our quantities are well-defined.
2) I can't see why Lagrangians are equivalent when their coefficients are changed. Sure, there is a well-defined procedure to change the Lagrangian; but you are changing it. If I choose my items (b) and (c), I still have to start with the Lagrangian adapted to (b) such that I get the correct results. It is cheating: I am not calculating something that was well-defined from the start, but I am adjusting the initial values in my calculation so as to get a correct result.
Also, I would say that one is usually not allowed to solve an equation of motion and to plug in the solution back into the Lagrangian. This is allowed only if you are solving an algebraic equation of motion (a holonomic constraint). For example, if \ddot x = y is one of your equations of motion, you can solve it for y, but if you substitute \ddot x instead of y into the Lagrangian, you get a Lagrangian with more derivatives than you had before, and the new equations of motion will not be equivalent to the old ones.
2. The point is that the theory is specified by an effective Lagrangian together with a cut-off, and that changing the cut-off prescription can be compensated by an adjustment of the coupling constants. As for solving for some variables, I meant the situation where the equations are such that one can actually do it in a valid manner. This is a classical analogue of changing the cut-off: lowering the cut-off is equivalent to "integrating out" some degrees of freedom, and in the classical limit this reduces to solving some equations of motion for these degrees of freedom.
1. To avoid ill-defined quantities, you should start with a Lagrangian together with a cut-off procedure and then show that the choice of the cut-off procedure is unimportant (can be absorbed into a finite adjustment of the coupling constants). Perhaps the following analogy will help. Faddeev and Popov originally explained how to define the path-integral for gauge theories by applying some formal manipulations (gauge-fixing) to an ill-defined path-integral. These manipulations require choosing a gauge condition and thus it is not clear that what one gets does not depend on this choice. A better way to explain this construction is to start with a gauge-fixed version of the path-integral and demonstrate that expectation values of observables quantities are unaffected by a choice of the gauge condition (this is usually done be means of BRST symmetry).
Quotation from T. Muta, "Foundations of quantum chromodynamics" - world scientific, 1987. Page 308.
" Though the coefficients f0, f1, f2, ... as well as the coupling constant g are scheme- dependent, the whole perturbative series (5.3.3) when summed to all orders is scheme-independent. In the practical application of perturbation theory, however, we truncate the series (5.3.3) at a certain order to obtain an approximate expression of the physical quantity f(g). Then the quantity f(g) itself turns out to be scheme-dependent in the neglected orders. This is the source of the renormalization-scheme dependence of perturbative predictions on the physical quantity. (For a review, see. e.g. [Hur 83].) It should be emphasized here that the renormalization-scheme dependence of the perturbative prediction on physical quantities is a general phenomenon not specific to quantum chromodynamics. The reason why we are concerned with this phenomenon particularly in quantum chromodynamics is that in QCD the renormalization-scheme dependence of the physical quantity in perturbation theory causes a non-negligible difference in the physical prediction since the coupling parameter g^2/(4 pi) is not sufficiently small and the size of the neglected orders is, in general, not very small. In the case of quantum electrodynamics the coupling parameter e^2/(4 pi) is sufficiently small for a very wide range of the renormalization scale and so the problem of renormalization-scheme dependence is practically irrelevant."
The series (5.3.3) is of course an asymptotic series, so there is no assurance that it can be summed and that the results are actually independent of the regularization scheme.
PS
Actually it seems that the beta functions are not measurable and thus can depend on the renormalization scheme.
I have designed an explanation of renormalizations available in Russian here http://fishers-in-the-snow.blogspot.com/2011/08/objasnenie-perenormirovok.html and in English here http://www.science20.com/qed_reformulation_feasible/blog/ultimate_explanation_renormalizations-81791 , and also in my blogs.
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branchandroot
(no subject)
Date: 2010-04-20 01:48 am (UTC)(no subject)
Date: 2010-04-20 01:50 am (UTC)(no subject)
Date: 2010-04-20 07:17 am (UTC)(no subject)
Date: 2010-04-20 02:36 pm (UTC)(no subject)
Date: 2010-04-20 02:57 pm (UTC)(no subject)
Date: 2010-04-20 03:15 pm (UTC)(no subject)
Date: 2010-04-20 03:33 pm (UTC)(no subject)
Date: 2010-04-21 02:10 pm (UTC)(no subject)
Date: 2010-10-03 12:44 am (UTC)(no subject)
Date: 2010-10-03 04:50 am (UTC)(no subject)
Date: 2010-10-03 11:18 am (UTC)Например, в алгебраической геометрии после Зарисского-ван дер Вардена-Вейля все было строго, а определения менялись, и совсем не ясно, что текущие определения - навсегда. Эта работа по продумыванию определений - не "предварительная". Равно она была не "предварительной" и в истории анализа. Анализ был до завершения (на данный момент) работы по продумыванию определений функций, рядов и т.п. и продолжает существовать и после, когда эти определения уже установились. Продумывание и поиск правильных определений - главное в развитии математики.
(no subject)
Date: 2010-04-20 05:29 pm (UTC)По-моему надъ этимъ вопросомъ сейчасъ почти никто вообще не работаетъ. А вопросъ важный, т.к. мы вообще не имѣемъ теоріи по-настоящему, пока не понимаемъ этого вопроса.
Напримѣръ, есть разные методы "перенормировки" (размѣрная, дзета-функція, Паули-Вилларсъ, и т.д.). Для нѣкоторыхъ теорій эти методы даютъ одинаковый отвѣтъ. Но Если бы разные методы "перенормировки" давали бы разные отвѣты, не согласующіеся съ экспериментомъ, то это вѣдь не означало бы, что теорія невѣрна - это означало бы, что мы просто не знаемъ, что дѣлаетъ данный методъ "перенормировки". Можетъ, есть какой-то другой методъ, дающій правильный отвѣтъ. Согласіе же отвѣта съ экспериментомъ опять-таки не значитъ, что методъ "перенормировки" правильный: есть много способовъ получать правильный отвѣтъ невѣрными вычисленіями. Пока мы не знаемъ, какой методъ по-настоящему правильный, мы не понимаемъ, что мы вычисляемъ.
(no subject)
Date: 2010-04-21 05:31 pm (UTC)I think this problem has been resolved quite satisfactorily in the 1960s, thanks to the works of Ken Wilson and his notion of effective action. I also think you are mixing two different things: regularization and renormalization. The 1st key point is that one should always think about QFT with a UV cutoff. Then there are no infinities, but there is a dependence on the details of the cut-off. The cut-off dependence is of two kinds: power-like and logarithmic. The 2nd key point is that this dependence can be compensated by the redefinition of the parameters of the Lagrangian. The 3rd key point is that logarithmic dependence of these parameters is controlled by the RG equations whose form can be computed with any cut-off. Different methods of regularization correspond to different choices of the cut-off, but it is a theorem that the resulting RG equations are independent of the choice, up to a redefinition of the coupling constants. Physical quantities (like S-matrix elements) therefore can be computed using any regularization method, and the results are guaranteed to be related by a redefinition of the coupling constants.
(no subject)
Date: 2010-04-21 10:14 pm (UTC)It seems to me that the problem has not been solved satisfactorily because things stand as follows with QFT (please correct me if I am wrong here):
1. QFT is formulated as a theory with a cutoff parameter. If I am to take this seriously then I must consider a Lagrangian with coupling constants that are functions of a parameter, which is not a measurable value but a theoretically introduced unknown number. Then I start computing something in perturbation theory using this Lagrangian. Sure enough, I get infinities. Then I realize that I need to cut off something. Mysteriously (and without mathematical justification), the same unknown number is to be used as a cutoff parameter in my perturbative calculations, so that I get finite answers instead of infinities. How can it be that the unknown parameter of the Lagrangian is the same number as the cutoff introduced by hand into some integral in the middle of my calculation? To me, this procedure just never made any sense. I can memorize this procedure, but I cannot understand what is being done. There must be an explanation of what we are calculating here, and then there will remain a technical question of how to calculate it.
2. There exist theories where actual results in perturbation theory (matrix elements or beta functions) do depend on the regularization scheme. Not in QED, but in some QCD calculations and also in some mock-ups of quantum gravity.
(no subject)
Date: 2010-04-21 11:13 pm (UTC)2. (a) The numerical values of the beta-functions are not physical. Beta-functions are components of a vector field on the space of coupling constants. Choosing a different renormalization scheme results in a reparameterization of the coupling constants, which in turns changes the numerical values of the components of the vector field. This does not mean that physical results are scheme-dependent. The vector field is still the same.
(b) Physical S-matrix elements are scheme-independent too. One sometimes decomposes them into unphysical "constituents". For example, one writes a hadron scattering cross-section in terms of parton distributions and parton scattering cross-sections. Parton distributions and cross-sections are scheme-dependent, but the hadron cross-section is scheme-independent.
(no subject)
Date: 2010-04-22 07:26 am (UTC)a) a Lagrangian with coupling constants depending on the cutoff parameter in a certain way,
b) a particular regularization scheme that will be later used for calculations, in which the same cutoff parameter also enters,
c) the value of the cutoff parameter.
Then one performs calculations and finds that the physical results are invariant under RG flow.
The RG flow corresponds to changing the value of the cutoff parameter, after which the objects a), b) also change simultaneously, so that the physical results stay the same.
I still have two uneasy questions about this:
1) We now have several specific methods of regularization: zeta-function, dimensional regularization, Pauli-Villars, and a few more perhaps that I can't remember now. We can perhaps prove the properties of the RG flow when these methods are used. How can one prove that any method of regularization invented in the future will still yield the same results?
2) If we change the method of regularization, e.g. if we change from zeta-function to dimensional regularization, then the functional dependence of the Lagrangian on the cutoff parameter will be different. It is not equivalent to changing the value of the cutoff parameter (this would be indeed just a different "gauge"). So, as far as I understand, the Lagrangian (a) needs to know in advance that we are using a particular regularization scheme (b) in the middle of our calculations. Or else there will be no cancellation of infinities.
If this is true then it appears that the theory is not really there. Imagine that we are doing classical electrodynamics where we need to solve a wave equation. We have two methods of solving the wave equation: say, by Fourier transform and by separation of variables. Of course, the results are the same: these are just two different techniques for solving a well-defined mathematical problem. Now, imagine that the results are incorrect unless you need to keep two different Lagrangians for the electromagnetic field: one Lagrangian is to be used when we solve the wave equation through the Fourier transformation, but a different Lagrangian is to be used when we solve the wave equation through separation of variables. The two Lagrangians are not equivalent and not simply a redefinition for convenience; these are two really different Lagrangians, with different coupling constants and maybe different terms.
Surely if this were the situation in classical electrodynamics, there would have been a major lack of understanding. If we really understand what we are calculating, then we should start with an initial well-defined mathematical object (e.g. a wave equation) and calculate something. It seems that in QFT we do not have such an initial mathematically well-defined object.
(no subject)
Date: 2010-04-22 04:16 pm (UTC)(1) The answer to the first question is that all methods of regularization, implicitly or explicitly, consist of modifying the theory in the UV by adding operators of arbitrarily high dimension to the action. That is, the difference between any two methods of regularization is the addition of such terms. Their effect on the low-energy physics (i.e. on the renormalizable terms in the low-energy effective action) is at most a finite change in the values of the dimensionless coupling constants and masses. Hence all methods are equivalent.
(2) The Lagrangians used with two different regularization schemes are the same. The only difference is that the functional dependence of the coupling constants on the cut-off is different. So the Lagrangians are related in a simple way: by a cut-off-dependent reparameterization of the coupling constants. There is a well-defined procedure how to express one set of coupling constants in terms of another one. How is this different, for example, from transforming one classical action to another one by solving equations of motion for some of the fields and plugging the solutions back in? The new action will look different, but it is still equivalent to the old one.
(no subject)
Date: 2010-04-22 09:44 pm (UTC)1) For example, there is the reasoning about UV modifications: one says that by adding operators to the Lagrangian the effect on the low-energy effective theory will be such and such. This would be a valid argument if we already had something well-defined that we are calculating, but we don't, and we are trying to use this very argument to show that our quantities are well-defined.
2) I can't see why Lagrangians are equivalent when their coefficients are changed. Sure, there is a well-defined procedure to change the Lagrangian; but you are changing it. If I choose my items (b) and (c), I still have to start with the Lagrangian adapted to (b) such that I get the correct results. It is cheating: I am not calculating something that was well-defined from the start, but I am adjusting the initial values in my calculation so as to get a correct result.
Also, I would say that one is usually not allowed to solve an equation of motion and to plug in the solution back into the Lagrangian. This is allowed only if you are solving an algebraic equation of motion (a holonomic constraint). For example, if \ddot x = y is one of your equations of motion, you can solve it for y, but if you substitute \ddot x instead of y into the Lagrangian, you get a Lagrangian with more derivatives than you had before, and the new equations of motion will not be equivalent to the old ones.
(no subject)
Date: 2010-04-23 12:26 am (UTC)1. To avoid ill-defined quantities, you should start with a Lagrangian together with a cut-off procedure and then show that the choice of the cut-off procedure is unimportant (can be absorbed into a finite adjustment of the coupling constants). Perhaps the following analogy will help. Faddeev and Popov originally explained how to define the path-integral for gauge theories by applying some formal manipulations (gauge-fixing) to an ill-defined path-integral. These manipulations require choosing a gauge condition and thus it is not clear that what one gets does not depend on this choice. A better way to explain this construction is to start with a gauge-fixed version of the path-integral and demonstrate that expectation values of observables quantities are unaffected by a choice of the gauge condition (this is usually done be means of BRST symmetry).
(no subject)
Date: 2010-04-21 10:30 pm (UTC)" Though the coefficients f0, f1, f2, ... as well as the coupling constant g are scheme-
dependent, the whole perturbative series (5.3.3) when summed to all orders is
scheme-independent. In the practical application of perturbation theory,
however, we truncate the series (5.3.3) at a certain order to obtain an
approximate expression of the physical quantity f(g). Then the quantity f(g)
itself turns out to be scheme-dependent in the neglected orders. This is the
source of the renormalization-scheme dependence of perturbative predictions
on the physical quantity. (For a review, see. e.g. [Hur 83].) It should be
emphasized here that the renormalization-scheme dependence of the
perturbative prediction on physical quantities is a general phenomenon not
specific to quantum chromodynamics. The reason why we are concerned with
this phenomenon particularly in quantum chromodynamics is that in QCD
the renormalization-scheme dependence of the physical quantity in
perturbation theory causes a non-negligible difference in the physical
prediction since the coupling parameter g^2/(4 pi) is not sufficiently small
and the size of the neglected orders is, in general, not very small. In the case of
quantum electrodynamics the coupling parameter e^2/(4 pi) is sufficiently
small for a very wide range of the renormalization scale and so the problem of
renormalization-scheme dependence is practically irrelevant."
The series (5.3.3) is of course an asymptotic series, so there is no assurance that it can be summed and that the results are actually independent of the regularization scheme.
PS
Actually it seems that the beta functions are not measurable and thus can depend on the renormalization scheme.
Renormalizations
Date: 2011-09-06 02:36 pm (UTC)