LESSWRONG
LW

[LINK] The Wrong Objections to the Many-Worlds Interpretation of Quantum Mechanics — LessWrong

25 [LINK] The Wrong Objections to the Many-Worlds Interpretation of Quantum Mechanics

by [anonymous]

19th Feb 2015

1 min read

25

Sean Carroll, physicist and proponent of Everettian Quantum Mechanics, has just posted a new article going over some of the common objections to EQM and why they are false. Of particular interest to us as rationalists:

Now, MWI certainly does predict the existence of a huge number of unobservable worlds. But it doesn’t postulate them. It derives them, from what it does postulate. And the actual postulates of the theory are quite simple indeed:

The world is described by a quantum state, which is an element of a kind of vector space known as Hilbert space.

The quantum state evolves through time in accordance with the Schrödinger equation, with some particular Hamiltonian.

That is, as they say, it. Notice you don’t see anything about worlds in there. The worlds are there whether you like it or not, sitting in Hilbert space, waiting to see whether they become actualized in the course of the evolution. Notice, also, that these postulates are eminently testable — indeed, even falsifiable! And once you make them (and you accept an appropriate “past hypothesis,” just as in statistical mechanics, and are considering a sufficiently richly-interacting system), the worlds happen automatically.

Given that, you can see why the objection is dispiritingly wrong-headed. You don’t hold it against a theory if it makes some predictions that can’t be tested. Every theory does that. You don’t object to general relativity because you can’t be absolutely sure that Einstein’s equation was holding true at some particular event a billion light years away. This distinction between what is postulated (which should be testable) and everything that is derived (which clearly need not be) seems pretty straightforward to me, but is a favorite thing for people to get confused about.

Very reminiscent of the quantum physics sequence here! I find that this distinction between number of entities and number of postulates is something that I need to remind people of all the time.

META: This is my first post; if I have done anything wrong, or could have done something better, please tell me!

Personal Blog

25

[LINK] The Wrong Objections to the Many-Worlds Interpretation of Quantum Mechanics

New Comment

102 comments, sorted by

top scoring

Click to highlight new comments since: Today at 3:12 AM

Some comments are truncated due to high volume. (⌘F to expand all)Change truncation settings

[-]JoshuaFox11y161

I'd like to see the best anti-MWI/Everett article out there.

5JoshuaFox11y

Here are the best items I have found in my search for anti-MWI reading. Some present anti-MWI arguments but in the end are pro-MWI. * David Wallace, The Emergent Multiverse (Pro; Anti-Everett arguments in the interludes) * Steven Weinberg: Lectures on Quantum Mechanics, sec. 3.7 (Seemingly pro-Everett, but in the end saying all current theories are flawed) * Adrian Kent, "Against Many-Worlds Interpretations" (Anti) * Stanford Encyclopedia of Philosophy "Many-Worlds Interpretation of Quantum Mechanics " (Mostly Pro, Anti in sec.6)

3BenLowell11y

Another relevant excerpt from the article:

3Rob Bensinger11y

It's not clear that 'MWI' in the LW sense has very much content -- in particular, it's not clear that Eliezer is committed when he endorses MWI to Carroll's description above. His recognition of the mysteriousness of 'reality fluid' and his willingness to entertain ideas like mangled worlds suggests he's perfectly happy to modify QM to generate the Born probabilities, provided the modifications reflect the character of physical law better than Bohmian Mechanics and objective collapse do. If MWI is just 'Bohmian mechanics is wrong' plus 'objective collapse is wrong' plus 'the right answer will structurally resemble our current physics', then the best articles criticizing MWI might be ones that defend Bohmian mechanics or an objective collapse theory as non-ridiculous, and argue that until we have a finished empirically adequate MWI we can't be confident it will actually turn out simpler or more-in-the-character-of-physics than BM.

[-]DanielLC11y100

I think MWI means more than that. If you figure out how to ensure that Schroedinger's cat is alive or dead, but not both, then it's not MWI. The mangled worlds thing gets rid of some of the worlds, but it most certainly does not get rid of all but one.

3JoshuaFox11y

Yes, those are all possibilities for what I am looking for. I'll let the experts decide: I'll be glad to read a coherent defense of Copenhagen, objective collapse, etc. or whatever it is that Hugh Everett/David Deutsch/Max Tegmark/Sean Carroll/etc are up against.

2n4r911y

You may be interested in (if you haven't already encountered) the "QBist" interpretation espoused by Fuchs, Mermin, Schack and others. Here are links to some appropriate papers by Fuchs, who in my opinion expresses the position most eloquently and efficiently: http://arxiv.org/abs/1003.5209 http://arxiv.org/abs/1311.5253 http://arxiv.org/abs/quant-ph/0205039 I personally see QBism as quite a natural extension of classical Bayesianism to quantum mechanics, and I am surprised that it is not discussed at all in this community. Given the interest that Less Wrong members have in quantum theory and its foundations, I can only surmise that this niche is due to some kind of idolization of Eliezer and his views. I am somewhat placated by your inclusion of Kent's paper in your list of coherent anti-MWI arguments, although I would love to see more of the genuine academic debate surrounding the interpretation and foundations of quantum theory faithfully reflected in this forum.

4Luke_A_Somers11y

Just from reading the abstract, I'm not interested in an egocentric model of the universe. Ontology doesn't follow the same rules as epistemology.

1n4r911y

QBism is less an egocentric model of the universe, more an egocentric interpretation of quantum theory. It doesn't say that we cannot have an ontological model of the universe; it says that quantum theory ain't it. However, I appreciate that this probably won't help with your lack of interest. Best of luck with everything.

0Luke_A_Somers11y

First, condescending snark? Seriously? Second, now I'm really confused. In what way can't QM be the fundamental ontology of the universe?

0n4r911y

I'm not sure how to interpret your question. If you're asking: "What is the case against the MWI interpretation of quantum theory?" then I would probably cite difficulties in explaining why our branch's history appears to be Born-rule typical as a major argument. If instead you're asking: "What is the case for a non-ontological interpretation of the wavefunction?" then the best I can do is attempt to summarise the arguments put forth in the above papers.

0Luke_A_Somers11y

I meant the second. If that is the point of the papers, then I guess that's fair enough, but, well, I don't anticipate that their argument is going to be valid. I'll go and read it; no need to summarize. First paper, I recognize that this is subjective and fuzzy, but... no? Bohm looks like an incredibly... well, I won't get into that, but it doesn't seem to me like a quick fix. Spontaneous collapse, okay, I grant that one. MWI doesn't seem like a quick fix in any sense either. No, that's silly. Yes, that was silly. But that's hardly the strongest argument that could or has been made. You don't get to pick your opponents' arguments like that. The way Decision theory is used when I've seen it is: any structure which is formally equivalent to a decider is a decider, and QM has such structures. ... No, that's stupid. No amount of sophistry can make predetermination relevant to the meaningfulness of 'decision'. And of course the relevance of this argument is dependent on taking the argument for probability-in-MWI to be strictly dependent on the applicability of decision theory in a particular way which is not the way it's actually being used. In particular, by the time you ascend the level of abstraction enough for decision theory to be relevant, you're past the point at which predetermination has fallen away and you're dealing with an effectively non-predetermined system. And then he just goes off and makes an argument that the theory is information about the state, not the state itself. But... ... ... if it successfully models your information about a thing, then the thing acts consistently with the model, which means you're also modeling the thing. There are theorems that constrain the ontology given these observations, and it basically boils down to 'QM is super legit'.

0n4r911y

I agree with you up until your last paragraph: the strength of Fuchs' papers are not in their direct criticism of Everettian interpretations (Kent's papers are a lot better at that). For your last paragraph, I think Fuchs would take umbrage at the idea that you are necessarily "modeling the thing" when you assign a quantum state to a given system. I don't think he believes that systems have a "true ontic state" of which quantum states are representative. Rather, the quantum state is merely a representation of an agent's beliefs about the future outcomes of their interventions/measurements into the universe. Nevertheless, Fuchs claims to be a scientific realist. I'm deliberately using the word "think" a lot here because I'm not confident of relaying Fuchs' views faithfully (this isn't directly my area of research). I haven't adopted a QBist interpretation (or any other), but from what I've read I feel it's worth serious discussion. You also mentioned theorems constraining ontology. You may be interested in Fuchs' take on Bell's Theorem: http://arxiv.org/pdf/1311.5253v1.pdf . I have been to a talk where he explains why the PBR theorem doesn't impact his interpretation, although the details evade me (and I can't find anything written about it by him online).

0TobyBartels11y

I'm also a fan of the Bayesian interpretation of quantum mechanics (and I've said so here a couple of times). I try not to say ‘Quantum Bayesianism’, because it seems to me that Fuchs has run with that term in directions that I don't necessarily want to go. (I'm an objective Bayesian, while Fuchs is a subjective Bayesian, and that's just the start.) Some fans of Everett avoid the term ‘many worlds’ and cringe at some of the writings of David Deutsch, for similar reasons. All of which is to say that if a hard-headed rationalist thinks that Fuchs is saying crazy things, run it against your model of a classical Bayesian saying similar things, and see if maybe it's the interpretation of Bayesianism that you object to rather than the interpretation of quantum mechanics, and adjust accordingly. (Of course, this doesn't help if you're not a Bayesian in the first place, but Bayes is more sacred here than Everett.)

0n4r911y

I don't know whether I'd consider myself a "fan" of any particular interpretation, but I think Quantum Bayesianism ranks highly in terms of the insight it sheds into the nature of quantum theory. I'd be interested in discussing or reading about the Bayesian interpretation in more detail, as I haven't had too much exposure besides Fuchs et al's papers and a couple of conference talks. For example, what is your take on the recent PBR theorem concerning the ontology of the quantum state, and would this depend on whether your Bayesianism is objective or subjective? Do you have any resources you'd particularly recommend?

1TobyBartels11y

My own brief summary of the subject is in an nLab article. (This is a math/physics wiki, and I assumed that the reader already knows quantum mechanics, at least up to the point of knowing what a density matrix is and what it's good for.) There are references there, but you'll notice that they're all linked from the History section. (Part of the point of that section is to make it clear that the idea predates Caves, Fuchs, et al, although they certainly deserve credit for making it prominent.) I don't know any over-all exposition that I really like, although I will always like the one cited as Baez 1993, which is where I learnt about it (and in fact where I first learnt about density matrices). That article doesn't say ‘Bayesian’, but as I was already a Bayesian when I read it, and since I knew Baez to be a Bayesian, I naturally interpreted it so. If you interpret the probabilities in a different way, then you'll get a very different interpretation of quantum mechanics as a result! Someday I want to write something for beginners, at the level of Eliezer's essays here (and in fact probably post it here too), but I haven't done that yet! Until then, Baez's piece is at the right level, but it doesn't address the things that LessWrongers specifically would want to see.

0n4r911y

I'd like to try and flesh out the difference between your personal interpretation and (for example) QBism. In your nLab article you describe an objective Bayesian is someone who "who naturally thinks of Bayesian probabilities as reflecting knowledge rather than belief, betting commitments, etc". This suggests that it has to be knowledge about something; about some objective ontological process I assume. Is this ontological process still somehow "quantum" in nature? Is it perhaps a hidden variable of some kind? You didn't reply to my previous question about the PBR theorem, which seems relevant since it places strong restrictions on hidden-variable-type psi-epistemic interpretations of quantum theory. I'd be very interested in hearing a response to that if you have the time.

0TobyBartels11y

Sorry, I forgot to answer you about PBR. I agree with Matt Leifer's analysis. Briefly: it's a fine theorem, and it's good that they proved it, but it shouldn't surprise anybody, and it doesn't rule out any of the interpretations that people actually advocate. As for my interpretation, I don't have any problems with Caves, Fuchs, and Schack's comprehensive 2001 paper on the subject (this is not their first 2001 paper, which was more about a technical result and vaguer on the interpretation). This paper writes extensively about states of knowledge. But since then, Fuchs has criticized that phrasing as insufficiently Bayesian (by which he really means insufficiently subjectivist). Quantum States: What the Hell Are They? at his website covers this, although it's hard to read. As you can see from the dates, he had these thoughts pretty early on. Anyway, if the original 2001 papers define the orthodoxy for the Bayesian interpretation, then I am an orthodox quantum Bayesian, and Fuchs is the heretic. Knowledge of what? Fuchs says knowledge of (or beliefs about, etc) the conesequences of one's interventions in a system; one can also say (which may be same thing) knowledge of the outcomes of further measurements. I would use more realist language: knowledge of the physical observables. If you try to build an ontological model in which each observable has an associated actual value and the results of measurements are determined by these values, then you'll have a hard time with that; but that's not what I want to do. An observable O does not (necessarily) have an actual value, but it has potential values (comprising its spectrum), and I have knowledge about O that can be summarized as a probability distribution over these potential values.

0n4r911y

To clarify: do you believe that there is something ontological in the system which is assigning probabilities of measurement outcomes in some way, when you make a measurement of the obervable O?

0TobyBartels11y

Probabilities aren't ontological; they're epistemological. I agree with everything that Eliezer writes about that, probabilities are in the map, etc. But remove that word; there is something ontological that assigns measurement outcomes when I make a measurement. Or to keep it simpler: when I make a measurement, the measurement outcome is ontological.

0n4r911y

A belated thank you for your replies. I feel like I'm starting to get the hang of what it means to take seriously the idea that probabilities are epistemological. It's difficult, moving between papers espousing differing interpretations, because their very language tends to presuppose some ontological commitment or other.

2TAG3y

There's a set of subjective/irrealist interpretations as well, which are routinely ignored here. Noticeably, Copenhagen is one such , but is constantly conflated with objective collapse here.

1TheAncientGeek11y

EY is sold on Deutschs presentation of MWI, which presents it as a kind of plain, unvarnished interpretation ... but that isn't a fact.. Others see MWI as making posits: that the wave function is, that there is state, that there is universal state, that there is a universal basis...

3[anonymous]11y

That's a good point; would anyone be interested in a follow-up sequence to the quantum physics sequence that presents arguments on all sides equally (at least, as equally as a biased individual can present them)? Eliezer sort of set up many worlds as the obvious hero and the Copenhagen interpretation as a weak enemy; there are more modern interpretations such as Quantum Bayesianism and pilot wave theory that are harder to dispel.

5DanielLC11y

Instructions for polls are here: http://wiki.lesswrong.com/wiki/Comment_formatting#Polls. In the mean time, I'll get this one for you. Would anyone be interested in a follow-up sequence to the quantum physics sequence that presents arguments on all sides equally? [pollid:823]

3Luke_A_Somers11y

Yes, contingent on finding a good author. Also, note that this poll is subject to heavy selection pressure based on who read the comments on this article.

2TheAncientGeek11y

It doesn't have to be one contributor, N authors could put the case for N alternatives to MWI.

1Angela11y

Assuming that the question means "would you be interested" and not "does there exist at least one person in the multiverse who would be interested".

1Luke_A_Somers11y

Yes, let's assume that. :|

1JoshuaFox11y

I'd be even more glad to read an article that specifically is against MWI/Everrett (or whatever you call it). David Wallace, The Emergent Multiverse, Interludes I and II, presents both sides but in the end is pro-MWI. A coherent, intelligent, reasonable article by an advocate of the other side would make things clearer.

0[anonymous]11y

Thanks, I completely forgot polls were a thing one could do.

1TobyBartels11y

While I agree, we shouldn't expect there to be any ‘the’ best anti-Everett article. There are any number of non-Everett positions, and they all disagree with Everett for different reasons. What we need are multiple articles putting forth the best arguments for various positions (or one large article with multiple sections). ETA: The main thing that this site really needs is just awareness that there is more to consider than MWI and objective collapse. I agree with Eliezer's arguments that MWI wipes the floor with objective collapse. That doesn't make me accept MWI, however. (Eliezer does do a short survey of other interpretations, but none of the ones that I think are worth consideration, besides Bohm's, show up.)

0MrMind11y

I like this

0gjm11y

That isn't an argument against MWI/Everett, it's an argument against thinking about quantum computation as "computing 2^n values of f(x) in parallel in different worlds". (On the grounds that what licences us to consider two Everett branches as separate "worlds" is that decoherence makes their interactions negligible, while what happens in quantum computation is all coherent and preventing decoherence is a central engineering problem. [EDITED to add: And that the operation of a "cluster state quantum computer" is not helpfully thought of in terms of doing lots of computations in parallel. I don't know enough about the cluster state model of quantum computation to have any idea how strong that argument is.])

[-]Viliam_Bur11y70

The comments below the article evoke my "Read the Sequences" emotions.

But where are those other universes?
Doesn't splitting the universe by making a decision contradict the conservation of energy?

And I feel like: "The conservation of energy is a rule within the universe; it does not apply to splitting universes. Even the notion of 'where' only applies within the universe. And the universe does not split 'when you make a decision'; it keeps splitting all the time regardless of the content of your neurons. Duh! Could we just skip this level of... (read more)

6JoshuaZ11y

One way of interpreting the math is even more basic: from the perspective of conservation of energy, there's still just one universe with different parts of the wave function not interacting.

3[anonymous]11y

I've daydreamed about debate clubs using kyu/dan levels like in go, but I don't think that would go over well with the majority of people. I think it would turn debating into a competition to memorize "standard" lines of argument so you can pass tests without actually understanding them fully. I can dream, though...

[-]Shmi11y60

I have commented there, and I will quote my comment here. To clarify, I am not anti-MWI, I am pro- experimental evidence.

It seems to me that you strawman a bit the main objection. Indeed, as you say

MWI certainly does predict the existence of a huge number of unobservable worlds. But it doesn’t postulate them. It derives them, from what it does postulate.

However, this does not answer the objection that

MWI is not a good theory because it’s not testable

if you phrase it the way the MWI opponents usually mean it:

MWI is no more testable than shut-up-an

... (read more)

4spxtr11y

He asserts that such an experiment exists. I would love it if he were to expand on this assertion.

2danieldewey11y

I don't have the expertise to evaluate it, but Brian Greene suggests this experiment.

1n4r911y

That experiment sounds very problematic to me. He says "After you measure the electron’s spin about the x-axis, have someone fully reverse the physical evolution.... Such reversal would be applied to everything: the electron, the equipment, and anything else that’s part of the experiment.". There is no explanation of the mechanics of how he thinks such a time-reversal could be implemented. We simply don't have the fine control over the quantum state of the entire measurement apparatus. In fact, the very assumption that quantum theory is even the true/applicable state of affairs at this macro scale is the kind of thing that many Copenhagenists dispute. Conversely, if it were possible to have such a fine control over the entire system including the very equipment used to perform the measurement, well then, you might as well simply make a quantum measurement of the larger quantum system which includes that apparatus! There would be different outcomes depending on whether collapse has or has not yet occured. It seems like whether or not this experiment even makes sense relies somewhat on whether MWI is true. Ultimately I think the very description of this experiment makes hidden assumptions, which beg the same question it is trying to answer.

1Shmi11y

Yes, dynamical collapse appears to make new falsifiable predictions. MWI doesn't, unless you take Deutsch's reversible quantum consciousness seriously.

0Aleksander11y

And even if you do, then the only viewpoint you will have really falsified is one which postulates that (a) the state vector collapse is caused by consciousness, and (b) concludes that therefore any consciousness has to do the trick, even one simulated on a quantum computer. I have met exactly zero physicists who'd treat (a) seriously, but even if you believe in (a), (b) still doesn't need to follow (someone could believe that only real human brain makes the magic happen). (I assume you were referring to experiment 3. from Deutsch's "Three experimental implications of the Everett interpretation in Quantum Concepts in Space and Time.")

1[anonymous]11y

Thanks for the reply; I will most likely be writing a longer, not-just-a-link post on some aspects of the argument that to my knowledge were not covered on Less Wrong. Namely, the experiments by David Deutsch which can distinguish between EQM and the Copenhagen Interpretation (but require quantum computers to get much, much better before we can test them), as well as arguments brought up by Sean Carroll and Scott Aaronson. As for your argument, the idea there aren't many worlds requires an extra collapse postulate, so many worlds is the default interpretation of quantum mechanics. The shut-up-and-calculate interpretation is just a refusal to accept any ontology whatsoever. I could easily turn it around and say; CI is no more testable than shut-up-and-calculate-only-probabilities-are-real, or than any other interpretation favored by a particular CI opponent, as long as that interpretation makes the exact same predictions as the orthodox QM. Since MWI and CI are on equal footing in that respect, and MWI has one less postulate, I think CI should be classified as a fringe position and everyone should provisionally accept MWI, unless they are a particularly dead-set on refusing any ontology at all. Of course, once we are capable of putting entire minds into superpositions we can just test it and find out once and for all.

4TheAncientGeek11y

That follows if you make two assumptions that are standard in the Sequences, but not elsewhere. One is that the Copenhagen Interpretation is the one that says collapse is a real physical process. Another is that MWI is the only non collapse theory.

1dxu11y

Are you arguing that it isn't?

3TheAncientGeek11y

Yes From wikipedia : According to Howard, wave function collapse is not mentioned in the writings of Bohr.[4]Some argue that the concept of the collapse of a "real" wave function was introduced by Heisenberg and later developed by John von Neumann in 1932.[23] However, Heisenberg spoke of the wavefunction as representing available knowledge of a system, and did not use the term "collapse" per se, but instead termed it "reduction" of the wavefunction to a new state representing the change in available knowledge which occurs once a particular phenomenon is registered by the apparatus (often called "measurement").[24] Note also that the objective function theories were put forward as novel theories, by people who were familiar with content of CI, at a much later period than the Bohr-Heisenberg discussions: the Penrose interpretation dates to the late 90s, GRW from 1985.

1Viliam_Bur11y

So the collapse only exists in straw-Copenhagen? Or is it yet another scientific theory that popularly gets mistaken for Copenhagen interpretation?

3TheAncientGeek11y

The CI was somewhat minimal interpretation, that posits some sort of transition between the quantum realm and classical information during measurement, but says little about its nature. That left the field open for other interpretations to be more specific about collapse/projection/measurement, variously portraying it as objective, subjective, actually decoherence, etc.

-1TheAncientGeek11y

If you have realism about the evolution of state according to the Schroedinger equation, then you need an extra collapse postulate to avoid many worlds.. But you don't have to have realism about state, so there are ways of avoiding many worlds without introducing collapse.

1TobyBartels11y

It would be very helpful if you would spell out these initialisms/acronyms.

2TheAncientGeek11y

Edited.

-7Shmi11y

0DanielLC11y

If your argument can be resolved by the scientific method, then why even bother arguing? You're often not that lucky. Should the government be bigger or smaller? In principle, you could randomly sort the 50 states into three groups where you increase the budget of one, decrease the budget of another, and leave the last as a control, but nobody's going to do that.

1Shmi11y

To come up with an argument which can be tested experimentally. At least in physics, which is the subject area here. Alas. Ah, but there are other ways. There are 200 other countries in the world, and some come very close to the experiments you want to perform, now or in the past. Does gun control reduce crime? Tested. Does universal healthcare reduce medical expenditures? Tested. Does changing attitudes toward women in STEM improve the fraction of exceptional women scientists? Tested. Does decriminalizing marijuana lead to more drug use? Tested. You can argue about these issues with all the Bayesian might until cows come home, it is no substitute for a good experiment.

7DanielLC11y

They let you look for correlations. They don't let you run a good experiment. People can come up with theories that explain the facts, just as MWI and the Copenhagen Interpretation explain the facts. Science won't save us here.

-1TheAncientGeek11y

To which the answer is "its not a theory , its an interpretation. Interpretations arent testable, they're philosophy. Including anti realism".

[-]PhilGoetz11y40

Some people will vote you down for posting something that's mostly just a link. Personally, I like good, relevant links.

How did you post this so as not to display its vote total?

[-]Shmi11y110

Votes are displayed only after a post is an hour or so old. Not sure what the actual value is. They are still shown on the sidebar.

9fezziwig11y

My understanding is that new posts don't show their vote totals right away, to help prevent snowball effects.

6Viliam_Bur11y

A link with an example from the article is much better than mere link. It gives you a good estimate of whether you want to read the article before you click the link.

[-]Jan_Rzymkowski11y10

Can anybody point me to what choice of interpretation changes? From what I understand it is an interpretation, so there is no difference in what Copenhagen/MWI predict and falsification isn't possible. But for some reason MWI seems to be highly esteemed in LW - why?

-1DanielLC11y

Because Copenhagen introduces additional rules that act in ways counter to everything we know about physics and gives no experimental evidence to justify them.

-4TheAncientGeek11y

Where Copenhagen means "Objective Reduction"

0dxu11y

What does this mean?

-2Shmi11y

Mostly because Eliezer wrote a number of highly emotional and convincing posts about it.

[-]Rukifellth11y00

Just to be clear, do these multiple-universes have the same qualities as the universe that we inhabit?

0DanielLC11y

Yes. It's possible that the vast majority of them have no life and we're in this one because of the anthropic principle, but beyond that they all act the same.

Moderation Log

Curated and popular this week

102Comments

102