# “cutoff” in an integral

Sometimes when physicists compute an integral (EDIT: in quantum field theory), they introduce what is called a "cutoff", which means an infinite bound in the integral is replaced by a finite parameter, calculations are made, and then the parameter that was introduced as a bound in the integral is taken out to infinity. My question is: how do you say "cutoff" in Russian in this context?

I guess the answer might be some prefixed form of сечение, but I was not successful in figuring this out from a web search. (I did discover in this way that катофф has been borrowed into Russian for use in poker, which must be of more recent vintage that the need to say "cutoff" in physics.)

Edit: If answers can be supported by pointing to a place where the translated term is used in a physics computational context, and which I might find without too much trouble, that would be appreciated.

• I am not sure, I properly understand what you are talking about. About solving an несобственный интеграл второго рода (improper intergral of the second kind) i. e. \int_a^b f(x) dx where f(x) \to \pm\infty in x = a or x = b? – Dmitry Alexandrov Jul 31 '14 at 12:08
• More or less, but procedurally we start with an integral over the whole real line or all of space, and figure it out by introducing a cutoff parameter L, integrate over |x| \leq L, and then at the end let L tend to infinity. – KCd Jul 31 '14 at 12:19
• Got an idea then. Usually such an integral has "бесконечный предел интегрирования", so in such cnotext you can say that you use/introduce "конечный предел (интегрирования) L" which you then tend to infinity. At least, I can imagine myself writing something like this. – Shady_arc Jul 31 '14 at 12:26
• I'm not looking for a word or phrase that will just happen to make sense, but rather the term that physicists really use. – KCd Jul 31 '14 at 12:29
• @DmitryAlexandrov: did those links to English examples help you? – KCd Aug 7 '14 at 4:30

Yes, they helped of course. I’ve googled around the Web and made sure that @vix2 is absolutely right: cutoff as a verb would be обрезать, as a noun – обрезание, though you may think that the latter sounds a bit funny (most common meaning of обрезание is circumcision :-). You can found it both in books / articles, e. g.:

Чтобы придать смысл расходящимся фейнмановским интегралам, вводится некоторая промежуточная регуляризация (ультрафиолетовое обрезание Λ).
[A. A. Славнов. Симметрии и перенормировка]

Поэтому любое ковариантоное обрезание (такое, как в методе Фейнмана) будет автоматически гарантировать исчезновение собственного натяжения...
[С. Швебер. Введение в релятивистскую квантовую теорию поля]

...and at least in one dictionary:

инфракрасное обрезание (ктп) — infrared cutoff
ультрафиолетовое обрезание (ктп) — ultraviolet cutoff
(‘ктп’ stands for квантовая теория поля)
[V. D. Novikov et al. The Russian-English Dictionary of Physics. — RUSSO, 2004]

However, I am still not sure that term is used enough widely since a) there are too few occurences in google; b) somewhere it is explained in parentheses via English cutoff.

Таким образом, это соотношение показывает что величину инфракрасного обрезания (IR cutoff) нельзя выбирать независимо от величины ультрафиолетового обрезания.
[Ускоренное расширение вселенной и голографический принцип]

But maybe the reason is that too few Russian quantum field theorist write in Russian or publish their preprints in the Internet.

Finally, may I ask why you need this?

• My interest was largely just curiosity, actually. – KCd Aug 7 '14 at 11:10
• I've looked up the stress on обрезание at gramota.ru and it says both обрЕзание and обрезАние are used. Do they have the same meaning, or is one stress usage the religious ritual and the other is everything else with the meaning of cutoff? – KCd Aug 7 '14 at 11:12
• @KCd That’s a good question. I always mark stressed syllables in discussing words here on R.SE, but now I’m just not sure. Common pronunciation is обреза́ние of course (as a proof see ‘Русское словесное ударение’ by Зарва), but there are quite a lot of professional dialectisms which differ from common words by stress. Since I had never heard обрезание in that context, I did not want to mislead you. But definitely there would be nothing wrong to say обреза́ние. – Dmitry Alexandrov Aug 7 '14 at 20:48
• @KCd Verbs обре́зать / обреза́ть are different grammatical forms: perfective and imperfective respectively. You know it, of course. – Dmitry Alexandrov Aug 7 '14 at 20:51
• @KCd You’d better ask vix2. He seems to be more familiar with QFT. – Dmitry Alexandrov Aug 7 '14 at 20:57

"Интеграл обрезается на высоких частотах в связи с конечными " "Интеграл обрезается на дебаевской моде" etc.

• Где ударение в слове обрезание, когда оно используется в физике? – KCd Aug 14 '14 at 12:17
• Интеграл обрезан, интеграл обрез'ается. – vix2 Aug 15 '14 at 21:16
• Спасибо, но я имел в виду обрезание: в физике оно обре́зание или обреза́ние? – KCd Aug 17 '14 at 16:15

"отсечка" works fine, I'd say there is not a good word for this concept for everyday use but "отсечка" works even if it sounds a bit like professional slang. The important thing is, you are going to be understood because this word means just that: some threshold after which you say "OK, enough, stop that and don't go any further"

In audio recording context I use "обрезание" for cutoff/clipping. When you are recording a signal that is too loud you don't actually want it to be cut off. Still the peaks get limited (note that I do not do this for life, just record now and then). So here rather than limiting value to a reasonable value you lose information by trimming the peaks of a signal.

"Сечение" means "section" (eg. conical section) and is a term with a completely different meaning: something you get when dissecting something else with a surface (often planar).

UPD: from a lecture here at page 12

Интегрирование обрывается на некотором значении x = b...

• Concerning your last remark, I had said that perhaps a prefixed form of сечение, not that word itself, might fit. For example, in math there is a concept called "truncated Witt vector", which in Russian is усеченный вектор Витта. Though in English the terms truncation and cutoff and not interchangeable in practice, so maybe it doesn't work that way in Russian (or maybe it does). – KCd Jul 31 '14 at 12:28
• Aah, regrading the regularisation parameter, it seems I thought too much into these. I have just re-read the article carefully. Basically, in quantum theory you call that Λ a regularization parameter because it is, well, a parameter which you use to perform regularization required to make a meaningful answer out of your divergent integrals. If what you do with your integral is not regularization then, naturally, there is no regularization parameter (just as you do not call any multiplier a "normalization constant" if you do not actually normalize anything) – Shady_arc Jul 31 '14 at 14:11

The most acceptable translation for Cut-off point would be точка отсечения. However, in the case it is common to say пренебрегаем вкладом в интеграл значений x > y`, поскольку [proof that ignored part is indeed small enough to be ignored].

The word you are looking for is Предел.

There is a wiki page about integrals: Определенный интеграл.

You can see there terms нижний предел and верхний предел meaning lower cutoff and upper cutoff respectively.

• Вы определенно невнимательно прочитали дискуссию в комментариях под вопросом. Да и сам вопрос, пожалуй, тоже. – Dmitry Alexandrov Aug 6 '14 at 2:00
• The term предел is not going to fit this context. The terms нижний предел and верхний предел for definite integrals are translated as lower bound and upper bound. – KCd Aug 7 '14 at 4:33