Are there any Jewish responses to the "Law of Attraction" or analyses of this idea in the context of Jewish thought?
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Ditto Alex's reference, "the rich get richer and the poor get poorer." Though that could simply be a comment on economics.
As far as birds of the feather flocking together, we find Bava Kamma 92b:
א"ל רבא לרבה בר מרי מנא הא מילתא דאמרי אינשי מטייל ואזיל דיקלא בישא גבי קינא דשרכי אמר ליה דבר זה כתוב בתורה שנוי בנביאים ומשולש בכתובים ותנן במתניתין ותנינא בברייתא כתוב בתורה דכתיב (בראשית כח) וילך עשו אל ישמעאל שנוי בנביאים דכתיב (שופטים יא) ויתלקטו אל יפתח אנשים רקים ויהיו עמו ומשולש בכתובים דכתיב [בן סירא יג] כל עוף למינו ישכון ובני אדם לדומה לו תנן במתני' כל המחובר לטמא טמא כל המחובר לטהור טהור ותנינא בברייתא רבי אליעזר אומר לא לחנם הלך זרזיר אצל עורב אלא מפני שהוא מינו
Said Rava to Rabba bar Mari, is there Torah support for the popular saying about [people hanging out with those similar to them]? He said: it's found in the Torah, repeated in the Prophets, found again in the Holy Writings [well sortof, if you count Ben Sira], again in the Mishnah, and again in the Braisa ...
Here's Rabbi Akiva's advice to his son (Pesachim 112a):
והוי משתדל עם אדם שהשעה משחקת לו
Get involved with a person whose fortunes are doing well
This could simply mean "if it looks like he's a good investor, invest with him", but I think it's also connected to the concept of mazal, roughly translated as "karma" (which is irrelevant to the course of the Jewish people as a whole, but still can matter to individuals). Again, it's an aggadic passage, so make of it what you will.
More on mazal (Horiyos 12a):
Maybe his mazal is down because he's depressed.
So try to keep your mood up. (Definitely better for your health and social life anyhow.)
So there's the concept of keeping yourself feeling positive, and that success can be contagious. Nowhere is there the notion (found in some disgustingly self-centered feel-good books today) that you should try to avoid hard-stricken people because their bad mazal is contagious; this runs contrary to our commandment to emulate G-d by being kind to people (and not just those people who are our best buddies and scratch our backs in return), including and especially those who need the most help.
Moving on to contemporary theology, I've heard those who say that if you have real faith, that will make something happen, the example is the guy who asks the rabbi, "so if I have 100% faith that this ticket will win the $1M lottery, it will win?" "Yes", says the rabbi, "so how about I give you half a million dollars for your ticket right now? If you thought about taking my offer, that proves you don't have perfect faith." This is somewhat troubling (to me, anyways) as it blurs the line between what's in our hands and what's in G-d's. See this piece and related Tradition Article (Rabbi Stein, 42:1, currently available for free) with the Chazon Ish making clear that we have faith that G-d can control any outcome, but not necessarily that it will be exactly how I want it. There's similar discussion with some motivational speakers I've heard talking about prayer, e.g. "prayer always works" [well it's a mitzva so it does something, but always what you want it to do?] it starts to sound like "if you pray hard enough you can force G-d" ... again troubling.
I think it's somewhat related to the Lubavitcher maxim, Tracht gut, ven zayn gut.
It may also be related to the popular adage "Poverty follows the poor," mentioned in the Gemara (Bava Kamma 92a, bottom, and in a couple of other places).
Look up the following Jewish sages, learn anything and everything they have to say about Bitachon (בטחון), and you will be light years ahead of the secular concept of "Law of Attraction".
-Rebbe Nachman of Breslev
-Madregas HaAdam, Alter of Novardok
-RamChal (Moshe Chaim Luzzato)
-Rav Yisrael Salanter
-Nefesh HaChaim, Rav Chaim Volozhiner
There are others, but the writings on Bitachon just from the list above should give you plenty of material for the next 120 years or so.